HIAS-E-23

Accuracy and Retaliation in Repeated Games
with Imperfect Private Monitoring:Experiments and Theory
Yutaka Kayaba
Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan

Hitoshi Matsushima
Department of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Tomohisa Toyama
Division of Liberal Arts, Kogakuin University, 1-24-2, Nishishinjuku, Shinjuku-ku, Tokyo 163-8677, Japan

March 18, 2016

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8604
http://hias.ad.hit-u.ac.jp/
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1

Accuracy and Retaliation
in Repeated Games with Imperfect Private Monitoring:Experiments and Theory 1
Yutaka Kayaba
Hitotsubashi University 2
Hitoshi Matsushima
University of Tokyo 3
Tomohisa Toyama 4
Kogakuin University
March 18, 2016
Current Version: April 14, 2016

Abstract
We experimentally examine repeated prisoners’ dilemma with random termination, where
monitoring is imperfect and private. Our estimation indicates that a significant proportion ofsubjects follow generous Tit-For-Tat (g-TFT) strategies, straightforward extensions of Tit-ForTat. However, the observed retaliating policies are inconsistent with the g-TFT equilibria.
Contrarily to the theory, subjects tend to retaliate more with high accuracy than with low accuracy.
They tend to retaliate more than the theory predicts with high accuracy, while they tend to retaliate
lesser with low accuracy. In order to describe these results as unique equilibrium, we demonstrate
an alternative theory that incorporates naïveté and reciprocity.
JEL Classification Numbers: C70, C71, C72, C73, D03.
Keywords: Repeated Prisoner’s Dilemma, Imperfect Private Monitoring, Experiments, Generous
Tit-For-Tat, Behavioral Model.

1

Matsushima, Toyama, and Yagi (2007), Matsushima and Toyama (2011), and Matsushima, Tanaka,
and Toyama (2013) are earlier versions of this paper. Research for this paper was financially supported
by a grant-in-aid for scientific research (KAKENHI 25285059) from the Japan Society for the
Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology(MEXT) of the Japanese government, and also by a research fund from the Center of Advanced
Research in Finance (CARF) at the University of Tokyo. We are grateful for Hide Ichimura and theparticipants in the seminars at SAET conference, Hitotsubashi University, and University of Tokyo for
their useful comments. We also appreciate Tomomi Tanaka and Nobuyuki Yagi for their helpful
supports. All errors are ours.
2 Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1 Naka, Kunitachi-city, Tokyo
186-8601, Japan. Email: yutaka.kayaba [at] r.hit-u.ac.jp
3 Department of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
E-mail: hitoshi [at] e.u-tokyo.ac.jp
4 Division of Liberal Arts, Kogakuin University, 1-24-2, Nishishinjuku, Shinjuku-ku, Tokyo 1638677, Japan. E-mail: tomohisa.toyama [at] gmail.com

2

1. Introduction
It is a widely accepted view that the long-run strategic interaction facilitates
collusion among players whose interests conflict with each other. The premise is that eachplayer observes information about which actions the opponents have selected before.
However, even if the monitoring of the opponents’ actions is imperfect (i.e. each player
cannot directly observe the opponents’ action choices but can observe informative
signals), still the theoretical studies have shown that sufficiently patient players canemploy cooperative strategies as equilibrium in a greater or lesser degree.
To be more precise, the folk theorem generally indicates that if the discount factor issufficient, i.e., close to unity, and each player can not directly but indirectly observe theopponents’ action choices through noisy signals, a wide variety of allocations can beattained by subgame perfect equilibria in the infinitely repeated game (Fudenberg, Levine,and Maskin (1994) and Sugaya (2012), for instance). Indeed the folk theorem isapplicable to a very wide range of strategic conflicts. However, the theorem does not
inform positively what kind of equilibria emerge empirically, as well as the strategies
people follow actually associated with the equilibria.
Given the lack of consensus on the strategies people empirically follow, this study
experimentally analyses our subjects’ behavior in the repeated prisoners’ dilemma. Our
experimental setup is imperfect monitoring. Each player cannot directly observe her
opponent’s action choice, but observes a signal instead, which is either a good signal or abad signal. The good (bad) signal is more likely to occur when the opponent selects thecooperative (defective) action rather than the defective (cooperative, respectively) action.
The probability that a player observes the good (bad) signal when the opponent selects
the cooperative (defective, respectively) action is referred as the monitoring accuracy,
which is denoted by p  ( 1 ,1) . The study experimentally controls the levels of2monitoring accuracy as treatments (the high accuracy pp

0.9 and the low accuracy

0.6 ) to examine the strategies our subjects follow. Specifically, the monitoring

technology is private in the manner that each player cannot receive any information about
what the opponent observes about the player’s choices (i.e., the signals are only
observable to the receivers).

3

To examine the prevalence of the strategies our subjects employ, following the recent
strand in the experimental repeated game in which the heterogeneity of the strategies
people follow is treated explicitly, we employ the Strategy Frequency Estimation Method(SFEM) developed by Dal Bó and Fréchette (2011). In the SFEM framework, we list
various strategies that our subjects potentially take, and then estimate the frequencies ofeach strategy. This list includes the strategies that share significant proportions in existingstudies of experimental repeated game, such as Tit-For-Tat (TFT), Grim-Trigger,
Lenience, and Long-Term Punishment. Distinctly from the existing experimental studies(Fudenberg, Rand, and Dreber (2012), Aoyagi, Bhaskar, and Fréchette (2015)), we
rigorously include stochastic strategies in our SFEM list. Importantly, we include
straightforward extensions of TFT, namely generous Tit-For-Tat (g-TFT).
Our SFEM estimates indicate that a significant proportion (about 70%) of our
subjects follow g-TFT strategies, though they follow heterogeneous g-TFT strategies. GTFT is a simple stochastic Markovian strategy that makes a player’s action choicecontingent only on the signal observed at the previous round and permits the stochastic
choice between the cooperative action and the defective action at each round. 5 Becauseof the permission of stochastic action choices, g-TFT has a great advantage to theequilibrium analysis over the TFT strategy: provided the discount factor is sufficient,
there always exist g-TFT equilibria, irrespective of the level of monitoring accuracy.
Motivated with such theoretical importance, we regard g-TFT strategies and equilibria as
the reasonable benchmarks of the standard repeated game theory with imperfect private
monitoring. Our estimates imply that g-TFT is well supported empirically as well as
theoretically.
Observing that many of our subjects follow g-TFT strategies, we empiricallyexamine their retaliation policies. We focus on the contrast of the probabilities ofcooperative action choices contingent on a good signal and a bad signal, which is referredas the retaliation intensity. 6 Fixing the discount factor sufficient, the retaliation
intensities are common across all g-TFT equilibria, depending on the level of monitoring
5

A deterministic version of g-TFT corresponds to the well-known Tit-For-Tat strategy with a slight
modification for the case of imperfect private monitoring, according to which, a player always mimics
her opponent’s action choice at the previous round by making the cooperative (defective) action choicewhenever she observes the good (bad, respectively) signal.
6 Note that TFT corresponds to the g-TFT whose retaliation intensity equals unity.

4

accuracy. The retaliation intensity implied by the g-TFT equilibria is decreasing in levelof monitoring accuracy. Importantly, this decreasing property plays the central role inmaking use of the improvement of monitoring technology and effectively saving thewelfare loss caused by the monitoring imperfection. Hence, it is quite important toexamine whether this property holds even empirically.
The retaliation intensities observed in our experimental data, however, contradict
with the predictions by the standard equilibrium theory mentioned above. Contrarily to
the g-TFT equilibria, our subjects tend to retaliate more in the high accuracy treatment
than in the lower accuracy treatment. They tend to retaliate more than the level impliedby the g-TFT equilibria in the high accuracy treatment, while they tend to retaliate lesser
in the low accuracy treatment. Hence, our subjects’ behavior cannot be explained by thestandard theory.
As a feedback from our experimental findings to the theoretical development, wedemonstrate an alternative theory that is more consistent with our experimental results
than the standard theory. This theory associates equilibrium behavior with psychology
and bounded rationality as follows. We permit each player to be motivated by not only
pure self-interest but also reciprocity. We permit each player to be often naïve enough toselect actions at random. We permit the degrees of such reciprocity and naiveté to be
dependent on the level of monitoring accuracy.
By incorporating reciprocity and naïveté into equilibrium analysis, we characterize
the underlying behavioral model of preferences that makes the retaliation intensity
implied by the g-TFT equilibrium increasing in level of monitoring accuracy, i.e., moreconsistent with our experimental results. In the contrast with the standard theory, thederived behavioral model guarantees the uniqueness of g-TFT equilibrium.
The organization of this paper is as follows. Section 2 reviews the literature. Section
3 shows the basic model. Section 4 introduces g-TFT strategy and equilibrium. Section 5
explains the experimental design. Section 6 shows the experimental results about
aggregate behavior. Section 7 explains the Strategy Frequency Estimation Method.
Section 8 shows the experimental results about individual strategies. Section 9
demonstrates the behavioral theory. Section 10 concludes.

5

2. Literature Review
This paper contributes to the long history of research on the repeated game literature.
The equilibrium theory demonstrates the folk theorems in various environments, which
commonly showed that a wide variety of outcomes are sustained by perfect equilibria
provided the discount factor is sufficient. Fudenberg and Maskin (1986) and Fudenberg,
Levine, and Maskin (1994) proved their folk theorems for perfect monitoring andimperfect public monitoring, respectively, where they utilized the self-generation natureof perfect equilibria explored by Abreu (1988) and Abreu, Pearce, and Stacchetti (1990),
which, however, crucially relies on the publicity of signal observations. For the study ofimperfect private monitoring, Ely and Valimaki (2002), Obara (2002), Piccione (2002)
explored the belief-free nature as an alternative to self-generation, which motivates aplayer to select both the cooperative action and the defective action at all times. They
showed the folk theorem in prisoners’ dilemma game where monitoring is private andalmost perfect. 7
Subsequently, Matsushima (2004) proved the folk theorem in prisoners’ dilemma
game with imperfect private monitoring by constructing review strategy equilibria as
lenient behavior with long-term punishments, where we permit the monitoring
technology to be arbitrarily inaccurate. Eventually, Sugaya (2012) proved the folk
theorem with imperfect private monitoring for a very general class of infinitely repeatedgames, by extending self-generation to imperfect private monitoring and then combining
it with the belief-free nature.
Matsushima (2013) is related to this paper, which intensively studied g-TFT
strategies in a class of prisoners’ dilemma games that includes this paper’s model 8. This
work corresponds to the theoretical benchmark of this paper, showing that the class of gTFT strategies has the great advantage of equilibrium analysis compared with TFT.
The literature of experimental studies on repeated games has examined thedeterminants of cooperation and tested various theoretical predictions, in order to findclues to resolve the multiplicity problem. See Dal Bó and Fréchette (2014) for a review.

7
8

For the survey up to almost perfect private monitoring, see Mailath and Samuelson (2006).
For other studies on g-TFT, see Nowak and Sigmund (1992) and Takahashi (1997).

6

The Strategy Frequency Estimation Method (SFEM), which is the methodology
employed in this study, is frequently used in the literature of experimental repeated games.
See Fudenberg, Rand, and Dreber (2012), Dal Bó and Fréchette (2011), Breitmoser(2015), Aoyagi, Bhaskar, and Fréchette (2015), for instance.
Importantly, this study includes g-TFT strategies and their variants in our SFEM list.
Inclusion of such stochastic action choices is scant in the literature of experimental
repeated game, with the exception of Fudenberg, Rand, and Dreber (2012). However,they include only a few g-TFT strategies just aiming to perform robustness checks fortheir main claims. 9 In contrast, we rigorously include many variants of g-TFT in our
SFEM list to examine our subjects’ retaliating policies.
The latter part of this paper shows a feedback from experiments to theory byincorporating behavioral aspects to rational behavior. This could be regarded as the firstattempt that provides the repeated game theory more relevancies to real behavior.

3. The Model
This paper investigates a repeated game played by two players, i.e., player 1 andplayer 2, in the discrete time horizon. This game has a finite round-length, but theterminating round is randomly determined, and therefore, unknown to players. The
component game of this repeated game is denoted by ( Si , ui )i{1,2} , where Si denotes
the set of all actions for player i  {1, 2} , si  Si , S { S1 u S 2 , s { ( s1 , s2 )  S ,ui : S o R , and ui ( s ) denotes the payoff for player i induced by action profile s  S
in the component game. We assume that each player i ' s payoff has the following
additively separable form:ui ( s )vi ( si )  wi ( s j ) for all s  S , where j z i .

The impact of a player’s action choice on her (or his) welfare is additively separable fromthe impact of the opponent’s action choice on her welfare.

9

The experimental setup in Fudenberg, Rand, and Dreber (2012) is imperfect public monitoring
where the stochastic strategies have relatively less importance than the case of imperfect private
monitoring.

7

At each round, two noisy signals Z1  :1 and Z2  : 2 occur after their action
choices are made, where :i denotes the set of possible Zi , Z { (Z1 , Z2 ) , and: { :1 u : 2 . A signal profile Z  : is randomly determined according to a conditional
probability function f ( | s ) : : o R , where¦ f (Z | s) 1 for all s  S , and weZ:

assume full support in that f (Z | s ) ! 0 for all Z  : and s  S .
Let fi (Zi | s ) { ¦ f (Z | s ) , and we assume that fi (Zi | s ) is independent of s j .
Z j : j

We denote fi (Zi | si ) instead of fi (Zi | s ) . We utilize Zi  :i to denote the signal forplayer ics action choice. Player i ' s action choice si influences the occurrence of thesignal for her action choice Zi , but does not influence the occurrence of the signal forthe opponent’s action choice Z j , where j z i .
We assume that monitoring is imperfect. At every round t  {1, 2,...} , each player icannot directly observe either the action, denoted by s j (t )  S j , that the opponent j z ihasselected,u ( s (t ))

or

even

the

realized

payoff(u1 ( s (t )), u2 ( s (t )))  R 2 , where we denote by s (t )profile,denotedby( s1 (t ), s2 (t ))  S the action

profile selected at round t . Instead, player i observes the signal for the opponent j ' s
action choice Z j (t )  : j , where j z i , through which, player i can not directly but
indirectly and imperfectly monitor the opponent j ' s action choice s j (t ) . 10
We assume that monitoring is private. Each player cannot know about what kind of
signal the opponent receives about her own action choice. Hence, each player i knows
her own action choice si (t ) and the signal for the opponent’s action choice Z j (t ) , butdoes not know either the opponent’s action choice s j (t ) or the signal for her own action
choice Zi (t ) .
The assumption of imperfect private monitoring, along with the assumption of the10

Our specification of monitoring structure is in contrast with the previous works such as Green andPorter (1984) and Aoyagi and Fréchette (2009). These studies commonly assumed that the distributionof a noisy signal depends on all players’ action choices, while we assume the above-mentioned
independence.

8

above-mentioned independence, eliminates the impact of a player’s observation about the
opponent’s observation about her action choices. This environment enables us to focus
on the impact of monitoring imperfection about the opponent’s action choices.

H

Let h(t )( s (W ), Z (W ))Wt 1 denote the history up to round t . Let us denote by{h(t ) | t

0,1,...} the set of possible histories, where h(0) implies the null history.

The average payoff for player i when the history h(t )  H up to round t occurs isdefined ast

U i (h(t )) {u ( s (W ))
¦
W
i

1

t

.

From additive separability, this average payoff can be rewritten as(1)

U i (h(t ))1[{W  {1,..., t}| si (W )tA} vi ( A)  {W  {1,..., t}| si (W )B} vi ( B)

 {W  {1,..., t}| s j (W )A} wi ( A)  {W  {1,..., t}| s j (W )B} wi ( B)] .

This paper specifies the component game as a prisoners’ dilemma with symmetry
and additive separability as follows. For each i  {1, 2} ,Si { A, B} , vi ( A) Y , vi ( B) 0 , wi ( A)wi ( B)X  Y , andX Y  Z ,where X , Y , and Z are positive integers, andZ ! Y ! 0.
This specification implies

u1 ( A, A) u2 ( A, A)

X,

u1 ( B, B) u2 ( B, B )X Y  Z ,u1 ( A, B) u2 ( B, A)X Z,

u1 ( B, A) u2 ( A, B )X Y .

andLet us call A and B the cooperative action and the defective action, respectively.
Selecting the cooperative action A instead of the defective action B costs Y , but9gives the opponent the benefit Z , which is greater than Y . Note that the payoff vector
induced by the cooperative action profile ( A, A) , i.e., ( X , X ) , maximizes the welfare

u1 ( s )  u2 ( s ) with respect to s  S , and is greater than the payoff vector induced by thedefective action profile ( B, B ) , i.e., ( X  Y  Z , X  Y  Z ) , because Z ! Y . Note alsothat the defective action profile ( B, B ) is a dominant strategy profile and the unique
Nash equilibrium in the component game.
We further specify the monitoring structure as:i{a, b} , fi (a | A)fi (b | B)

p , and 1  p  1 .
2

Let us call a and b the good signal and the bad signal, respectively. The probability
index p implies the level of monitoring accuracy. The greater p is, the moreaccurately each player can monitor the opponent’s action choice. The inequality p ! 1

2

implies that the probability of the good signal for a player occurring is greater when thisplayer selects A than when she selects B .
For each history h(t )  H up to round t , we define the frequency of cooperative
action choice A , or, the cooperation rate, by

U ( h(t )) {{W  {1,..., t} | S1 (W )A}  {W  {1,..., t} | S2 (W )
2t

A}

.

From (1) and symmetry, it follows that the sum of the average payoffs, i.e., the welfare in
average, when the history h(t )  H up to round t occurs is given by(2)

U1 (h(t ))  U 2 (h(t ))2[ U (h(t )){v1 ( A)  w1 ( A)}  {1  U (h(t ))}{v1 ( B)  w1 ( B)}]
2[ U (h(t )) X  {1  U (h(t ))}( X  Y  Z )]2[ X  {1  U (h(t ))}(Y  Z )] .

From (2), the cooperation rate

U (h(t ))uniquely determines the welfare

U1 (h(t ))  U 2 (h(t )) .
We assume constant random termination, where G  (0,1) denotes the probabilityof the repeated game continuing at the end of each round t , provided this game continues
up to round t  1 . Hence, the repeated game is terminated at the end of each round t t 1

10with a probability G t 1 (1  G ) . The expected length of the repeated game is given by
f

¦G

t 1

(1  G )t .

t 1

Throughout our experiments, we assume thatG

0.967 (=29/30),

implying that the continuation probability is very high, which mimics the discount factor
that is sufficiently large to support the existence of equilibria in which players colludewith each other in infinitely repeated interactions.

4. Generous Tit-For-Tat Equilibrium
Let D i  [0,1] denote a stochastic action choice for player i , implying theprobability that she makes the cooperative action choice A . Player i ' s strategy in therepeated game is defined as V i : H o [0,1] . According to V i , she selects the cooperative
action A with the probability of V i (h(t  1)) at each round t , provided the historyh(t  1) up to round t  1 occurs. Let 6i denote the set of all strategies for player i .

Let V { (V 1 , V 2 ) and 6 { 61 u 6 2 .
The expected payoff for player i induced by V  6 when the level of monitoring
accuracy is given by p  (0,1) is defined asf

U i (V ; p ) { (1  G ) E[¦ G W 1ui ( s (W )) | V , p] ,W 1

or, equivalently,f

U i (V ; p )tE[¦ G t 1 (1  G )¦ ui ( s (W )) | V , p ]
t 1

f

¦G

W 1

t 1

(1  G )tt 1

fE[¦ G t 1 (1  G )tU i (h(t )) | V , p ]
t 1

f

¦G

t 1

,

(1  G )tt 1

where E[ | V , p ] denotes the expectation operator conditional on (V , p ) . The expected

11frequency of cooperative action choice A , i.e., the expected cooperation rate, induced
by V  6 when the monitoring accuracy is given by p  (0,1) is defined asf

U (V ; p ) {E [¦ G t 1 (1  G )t U ( h (t )) | V , p ]
t 1

f

¦G

t 1

.

(1  G )tt 1

From additive separability and (2), it follows thatf

U1 (V ; p )  U 2 (V ; p )E[¦ G t 1 (1  G )t{U1 (h(t ))  U 2 (h(t ))}| V , p]t 1f

¦ G (1  G )t
t 1

t 1

2[ X  {1  U (V ; p )}(Y  Z )] .

Hence, the expected cooperation rate U (V ; p ) uniquely determines the sum of the
expected payoffs, i.e., the welfare in expectation U1 (V ; p )  U 2 (V ; p ) . A strategy profileV  6 is said to be a (subgame perfect) equilibrium in the repeated game with monitoring
accuracy p  (0,1) if
U i (V ; p ) t U i (V ic, V j ; p ) for all i  {1, 2} and all V ic  6i . 11

We introduce a class of strategies, namely, generous Tit-For-Tat strategies, as
follows. A strategy V i  6i is said to be generous Tit-For-Tat (g-TFT) if there exists(q, r (a), r (b))  [0,1]3 such thatr (a) ! 0 ,V i (h(0)) q ,for every t t 2 and h(t  1)  H ,V i (h(t  1)) r (a) if Z j (t  1) a ,andV i (h(t  1)) r (b) if Z j (t  1) b .
At round 1, player i makes the cooperative action choice A with the probability of q .
At each round t t 2 , player i makes the cooperative action choice A with the
11

The full-support assumption makes the distinction between Nash equilibrium and subgame perfectNash equilibrium redundant.

12

probability of r (Z j ) when she observes the signal Z j (t  1) Z j for the opponent’s
action choice at round t  1 . We will simply write (q, r (a ), r (b)) instead of V i for any
g-TFT strategy.
A g-TFT strategy (q, r (a ), r (b)) is said to be an equilibrium in the repeated gamewith accuracy p  (0,1) if the corresponding symmetric g-TFT strategy profile is anequilibrium in the repeated game with accuracy p . Let us define(3)w( p ) {Y.
G (2 p  1) Z

Note that0  w( p ) d 1 if and only if G t

Y.
(2 p  1) Z

The following theorem shows that a g-TFT strategy (q, r (a ), r (b)) is an equilibrium if
and only if the difference in cooperation rate between the good signal and the bad signal,i.e., r (a )  r (b) , is equal to w( p) .

Theorem 1: A g-TFT strategy (q, r (a ), r (b)) is an equilibrium in the repeated game withaccuracy p if and only if
(4)

Gt

Y
,
(2 p  1) Z

and
(5)

r (a )  r (b)Proof: Selecting siw( p ) .

A instead of B costs player i Y at the current round, whereas

at the next round, she (or he) can gain Z from the opponent’s response with probability
pr (a )  (1  p )r (b) instead of (1  p )r (a )  pr (b) . Since she must be incentivized to

select both action A and action B , the indifference between these action choices must
be a necessary and sufficient condition, i.e.,Y  G Z { pr (a )  (1  p )r (b)} G Z {(1  p )r (a )  pr (b)} ,that is,Y

G Z (2 p  1){r (a )  r (b)} ,13

implying (5). Since r (a )  r (b) d 1 , 1 tY, i.e., the inequality (4), must hold.
G (2 p  1) Z

Q.E.D.

From Theorem 1, whenever G t

Y, by setting r (a )  r (b)(2 p  1) Z

w( p ) , we canalways construct a g-TFT equilibrium. This implies the advantage of the investigation of
g-TFT strategies instead of the TFT strategy in equilibrium analyses, because the profileof the TFT strategies is not an equilibrium as long as G zequilibrium always exists whenever G t

Y, while g-TFT(2 p  1) ZY
. 12
(2 p  1) Z

This paper regards the observed difference in cooperation rate between the goodsignal and the bad signal as the intensity that subjects retaliate against their opponents.
We name it the retaliation intensity. Note from Theorem 1 that if subjects play a g-TFT
equilibrium, then the resultant retaliation intensity is approximately equal to w( p ) .
Importantly, the retaliation intensity implied by the g-TFT equilibria w( p ) is
decreasing in p . The less accurate the monitoring technology is, the more severely
players retaliate against their opponents. This decreasing property is essential forunderstanding how players overcome the difficulty of achieving cooperation underimperfect private monitoring.
In order to incentivize a player to make the cooperative action choice, it is necessarythat her opponent makes the defective action choice when observing the bad signal more
often than when observing the good signal. In other words, the retaliation intensity must
be positive. When monitoring is inaccurate, it is hard for her opponent to detect whether
the player actually makes the cooperative action choice or the defective action choice. In
this case, the enhancement in retaliation intensity is necessitating for incentivizing theNote that the g-TFT equilibrium does not depend on the assumption that Z1 and Z2 are
independently distributed. Whether a g-TFT strategy is equilibrium or not is irrelevant to whether
monitoring is private or public. For instance, any g-TFT strategy that satisfies (4) and (5) is anequilibrium even if each player can observe both Z1 and Z2 , i.e., even under imperfect public
monitoring.
12

14

player. Hence, the retaliation intensity must be decreasing in the level of monitoring
accuracy.
This decreasing property plays the central role in improving the welfare by utilizing
the noisy signals as much as possible. Since monitoring is imperfect, it is inevitable thatthe opponent observes the bad signal even if the player actually makes the cooperative
action choice. This inevitably causes the welfare loss to occur, because the opponent
might retaliate against the player even when the player actually makes the cooperative
action choice. In this case, if the monitoring technology is more accurate, the opponent
can well incentivize the player with being less sensitive to whether the observed signal isgood or bad, safely making the retaliation intensity lower. This serves to decrease thewelfare loss caused by the monitoring imperfection. Hence, it is very crucial in welfare
consideration to examine whether the experimental results satisfies this decreasing
property.
This study experimentally evaluates the retaliation intensities. We will regard g-TFT
strategies as the most appropriate theoretical benchmark. G-TFT strategies are thestraight-forward stochastic extensions of the well-known Tit-For-Tat strategy, extended
to have simple subgame perfect equilibria in imperfect private monitoring. G-TFT
strategies often appear as the canonical strategies in the theoretical development ofequilibrium with private monitoring. Not merely is the existence of the equilibria
guaranteed, but also, g-TFT strategies are the only equilibrium strategies that provide thesimple, analytically tractable characterization. The analytical tractability is appealing in
the empirical investigation of retaliation intensities.

5. Experimental Design
We conducted four sessions of the computer-based laboratory experiments 13 at theCenter for Advanced Research for Finance (CARF), University of Tokyo in October 2006.
We recruited 108 subjects in total from the subject pool consisting of undergraduate andgraduate students in various fields. Our subjects were motivated with monetary
incentives; the earned points in the experiments were converted into Japanese Yen with a
13

The experiment was programmed and conducted with the software z-Tree (Fischbacher (2007)).

15

fixed rate (0.6 JPY per point). Additionally our subjects are paid a fixed participation fee,
which is 1,500 JPY.
As demonstrated in Section 3, to make the structure of the game simplest, we adopt
the prisoners’ dilemma with symmetry and additive separability for our component game,in which payoff parameters are characterized as ( X , Y , Z )(60,10,55) . The payoffparameters have a structure in which the cost for cooperation Y is small so that the gTFT equilibria exist even if the monitoring technology is poor. The payoff matrix
employed in the experiment is displayed in Table 1. The labels on the actions and signals
are presented in neutral language (i.e. the actions are labeled “A” and “B” instead of“cooperation” and “defection”, and the signals are labeled “a” and “b” instead of “good”
and “bad”).
[TABLE 1 HERE]
The experiments have two treatments on the monitoring accuracy; one treatment is
the case in which the monitoring accuracy is high. In this treatment, the signals the player
observes and the action choices by the opponent coincide with 90% chance ( p0.9 ),and mismatch with 10% chance. We refer this treatment as “high accuracy treatment”.
The other treatment is the case in which the monitoring technology is poorer. The signals
match with the opponent’s action choices with only 60% chance ( p0.6 ), which isslightly larger than the chance level (50%). We refer this treatment as “low accuracy
treatment”.
All subjects in the four sessions received both treatments, but the treatment order iscounter-balanced to minimize order effects. First two sessions (52 subjects) started withthree repeated games of the low accuracy treatment then proceeded to three repeatedgames of the high accuracy treatments. Second two sessions (56 subjects) are conductedin the reverse order, which started with the high accuracy treatment then proceeded to the
low accuracy treatment. Each treatment is preceded with a short practice repeated game
which consists of two rounds, to let the subjects understand the new treatment. Subjects
were randomly paired at the start of each repeated game, and the pairs remained
unchanged until the end of the repeated game. The summary of the treatment order,

16

number of subjects, game lengths in repeated games which are determined by thecontinuation probability that is explained soon, are displayed in Table 2.
[TABLE 2 HERE]
We employ Constant Random Termination in our experiment to mimic the infinitely
repeated interactions. Each repeated game is to be terminated at each round with aconstant probability. We let the continuation probability be G

0.967( 29 / 30) .

With the probability of 1  G ( 1/ 30) , the repeated game was terminated at thecurrent round, and subjects were re-matched with a new opponent and proceeded to thenext repeated game. Indeed our subjects were not informed in advance which round is the
final round for each repeated game, otherwise we were not able to mimic infinitely
repeated games due to “the shadow of the future” (Dal Bó (2005)). To help our subjects
understand that the termination rule is stochastic and the probability of termination ateach round is 1/30, we presented 30 cells (numbered from 1 to 30) in the computer screen
at the end of each round; one number is selected randomly at the end of each round, andthe repeated game is terminated if the number 30 is selected by chance and the 30th cellin the computer screen turns to green, otherwise all the cells numbered 1 to 29 turns togreen at once and the repeated game continues. The screen is demonstrated in Figure A.1
and A.2 in Appendix 1.
Each subject is informed of the rule of games demonstrated above and the ways inwhich the game is processed in the computer screen with the aid of printed experimental
instructions. The instruction is also explained aloud by a recorded voice. Moreover, on
the computer screen during the experiments, our subjects were always able to look overthe structure of the game (the payoff matrix and the accuracy of the signals in thetreatment), the history up to the round in the repeated game, that consists of the actionsof her own and the signals on the opponent’s actions. 14

6. Aggregate Level Analysis of Experimental Results
14

See Appendix 6 for the experimental instructions and the images of the computer screen, translated
in English from the original Japanese version.

17

6.1. Overall Cooperation Rates and Round 1 Cooperation Rates[TABLE 3 HERE]
Table 3 displays the descriptive summary of the data. The entire 108 subjects made
8,864 decisions in the high accuracy treatment, and 9,144 in the low accuracy treatment.
The overall frequency of cooperative choices, i.e., the cooperation rate, is 0.672 in thehigh accuracy treatment, and 0.355 in the low accuracy treatment. The former isstatistically significantly larger than the latter (p < 0.001, Wilcoxon matched-pair test forindividual-level means). The first look of the cooperation rates suggests that our subjects
cooperate more as the monitoring technology is improved. As shown in Section 4, thecooperation rate uniquely determine the welfare of the players in our payoff setting (theadditive separable payoff structure). The larger cooperation rate in the high accuracy
treatment indicates that the welfare is improved in the high accuracy treatment.
To examine our subjects’ attitude for cooperation more directly, we focus on round1 cooperation rates (the frequency of cooperative action choices at round 1). The round 1
cooperation rates are one of the primary measures that directly reflect subjects’ motivation
for cooperation under the structure of the game (i.e. payoff parameters, discount factor,
and most importantly in our study, monitoring accuracy), independently from the historyof the incoming signals in the latter rounds, and also independently from the behavior of
the opponent matched in the repeated game.
[TABLE 4 HERE]
Table 4 presents the frequency of cooperation at round 1. The round 1 cooperation
rate is 0.781 in the high accuracy treatment, and 0.438 in the low accuracy treatment,
which is significantly smaller (p < 0.001) than the former. Our subjects tend to start
repeated games with cooperative action choices in the high accuracy treatment, however,their motivation for cooperation is discouraged as the noise in the signal is increased, tothe extent that they start with cooperative actions with less than 50% chance. This result

18

is somewhat surprising given that the experimental parameters (i.e. payoff parameters anddiscount factor) are conductive to the g-TFT cooperative equilibria even in the low
accuracy treatment. Despite the favorable environment for cooperation, our results
indicate that, from the start, non-negligible number of our subjects turn to be reluctant tocooperate in the low accuracy treatment.
These results also imply that our subjects differentiate their strategies between the
two treatments. Reacting to the change of the signal quality, perhaps they switch theirstrategies from cooperative ones in the high accuracy treatment to less cooperative onesin the low accuracy treatment. We explore the specific strategies our subjects follow ineach treatment in Section 9.

6.2. Signal-Contingent Cooperation Rates
Table 4 also displays signal-contingent cooperation rates. The frequency ofcooperative actions after observing a good signal, denoted by r (a; p ) , computed as thesimple mean of entire choices, is 0.852. We also report the alternative value which is themean of individual-level means, concerned with the possibility that the behavior of thecooperative subjects might be over-represented in the simple mean of choices. 15 The
mean of the individual-level means is 0.788, which is smaller by 0.064 than the simple
mean of choices, which implies that the over-representation might be the case. However,both two measures are consistently high and reaching around 0.8, indicating that our
subjects are quite cooperative when they observe a good signal in the high accuracy
treatment.
In the low accuracy treatment, the cooperation rate after observing a good signal is
not as high as observed in the high accuracy treatment. The simple mean of thecooperative choices is 0.437, and the mean of individual-level means is 0.423. Both low
values in the cooperation rates around 0.43 indicate that, similarly to the case of the round1 cooperation rate, our subjects are reluctant to cooperate even after observing goodsignals in the low accuracy treatment, opposed to the case in the high accuracy treatment.
15

Since the subjects who are cooperative might observe good signals more often and take cooperative
actions more, their cooperative choices might be over-represented in the computation of cooperation
rates contingent on a good signal.

19

The direct comparison of the cooperation rates between the two treatments indicates thatthe cooperation rate in the high accuracy treatment is larger than that in the low accuracy
treatment (p < 0.001 for both the simple mean and the mean of individual-level means).
As to the case of the cooperation rates after observing a bad signal, denoted byr (b; p ) , the simple mean of cooperative choices in the high accuracy treatment is 0.344.

Again, we also report the mean of individual-level means concerned with the overrepresentation of the subjects who tend to retaliate more. The value is 0.443, which islarger than the simple mean of choices by 0.104, demonstrating that the concern might be
the case here. However, still both measures are below 50%, indicating that our subjects
tend to defect rather than cooperate after observing a bad signal in the high accuracy
treatment.
This tendency of our subjects taking defections after observing a bad signal is more
apparent in the low accuracy treatment. The simple mean of cooperative actions over theentire choices is 0.272 and the mean of the individual-level means is 0.279, which areconsistently smaller than those in the high accuracy treatment (p < 0.001 for both of themeans). Observing a bad signal, our subjects tend to take more defections in the low
accuracy treatment than in the high accuracy treatment.
The overall picture on the round 1 cooperation rates and the signal-contingent
cooperation rates shown above robustly demonstrate that our subjects take morecooperative actions under the better signal quality, irrespective of signals they observe.
RESULT 1-a: Our subjects tend to cooperate more in the high accuracy treatment thanin the low accuracy treatment, adapting their strategies according to the signal accuracy.

6.3. Retaliation Intensity[TABLE 5 HERE]
Now we focus on the retaliation intensities which is one of our primary concern of
the study. We examine whether the observed retaliation intensities coincide with thevalues implied by the g-TFT equilibria ( w( p ) ). Table 5 presents the retaliation intensities

20

in the aggregate level. In the high accuracy treatment, the retaliation intensityr (a;0.9)  r (b; 0.9) is 0.508 in the mean of entire choices and 0.352 in the mean ofindividual-level means. Again, the discrepancy might be due to over-representation.
However, both measures of the retaliation intensity consistently differ from zerostatistically significantly (p < 0.001 for both measures). These results indicate that our
subjects use signal-contingent information in their action choices, perhaps attempting toincentivize the opponent to cooperate.
In comparison to the theoretical values implied by the g-TFT equilibria, bothmeasures of the retaliation intensity in the high accuracy treatment are statisticallysignificantly larger than the theoretical values ( w(0.9) 0.235 , p < 0.001 for bothmeasures). Thus, empirically, our subjects tend to rely on stronger punishments which are
more than enough to incentivize the opponents to take cooperative actions in the highaccuracy treatment.
On the other hand in the low accuracy treatment, the observed retaliation intensityr (a;0.6)  r (b; 0.6) is 0.165 in the simple mean of entire choices, and 0.144 in the meanof individual-level means. The two measures consistently differ from zero significantly(p < 0.001 for both measures), which demonstrates that, similarly to the case of the highaccuracy treatment, our subjects also use the signal-contingent information even in thepoorer monitoring technology. However, unlike the case of the high accuracy treatment,
the observed retaliation intensity in the low accuracy treatment is smaller than the level
implied by the g-TFT equilibria. Both measures of the retaliation intensity are
outstandingly smaller than the theoretically implied value w(0.6) 0.94 (both for p <0.001). Although our subjects do retaliate according to the signals in the low accuracy
treatment, however, the strength of the retaliation is far below the level enough toincentivize the opponents to cooperate, allowing the opponents to defect permanently to
pursue larger payoffs. 16

16

One might concern that the seemingly weaker retaliation intensities observed here do notnecessarily imply weak retaliation policies of our subjects, since our subjects might use long-term(multi-round) punishments; long-term punishing strategies punish opponents even observing a goodsignal during punishing phases, which could lower the retaliation intensity computed here. However,
our analysis in Section 8 indicates that only a marginal number of our subjects could take long-term
punishing strategies, hence the effect by long-term punishments is minimum.

21

The retaliation intensities are not only inconsistent with the values implied by the gTFT equilibria, but also the deviation is systematic. The direct comparison of theretaliation intensities between the two treatments indicates that the retaliation intensity inthe high accuracy treatment is larger than that in the low accuracy treatment (p < 0.001).
This is diametrically opposite to the implication by the g-TFT equilibria in which people
should employ weaker retaliating policies in the high accuracy treatment than in the low
accuracy treatment.
RESULT 2-a: Our subjects tend to retaliate more than the level implied by the g-TFT
equilibria in the high accuracy treatment, while they tend to retaliate weaker in the low
accuracy treatments. Moreover, opposed to the implications by the standard theory, theretaliation intensity is larger in the improved monitoring technology.

6.4. Reliance on Long Memories
In relation to the behavioral difference of our subjects along with signal qualities, it
is also interesting to examine whether our subjects tend to rely on longer histories ofsignals (i.e. signals two periods ago). Given the theories in review strategies (Radner(1986), Matsushima (2004), and Sugaya (2012)), one presumably assumes that people
rely on signals in longer histories to compensate the informational disadvantages inpoorer monitoring technologies.
To test whether our subjects rely on information in a signal occurring two periods
ago, here we fit the data with a probabilistic linear regression model, regressing the action
choices (a dummy variable which takes one if our subjects play cooperation) on all
memory-1 histories which consist of a signal and an action at the previous round, andfurther include information of a signal two period ago into the set of the explanatory
variables to the extent that the non-singularity holds without intercept. 17 The regression
coefficients on the signal two period ago capture the additional impact on cooperation
probabilities from the signal. The standard error is computed by a cluster-bootstrap

17

We borrow this approach in Breitmoser (2015).

22(subject and repeated game level) to control subject heterogeneity, which is also used tocompute the p-value.
[TABLE 6 HERE]
Table 6 displays the regression results. Opposed to the speculation above, theregression coefficients on the information two period ago are exclusively significant onlyin the high accuracy treatment, and none is significant in the low accuracy treatment.
Even in the high accuracy treatment, the size of the coefficients are only marginal, as the
maximum impact on cooperative choices is only 13.2%.
RESULT 3-a: There is no evidence that our subjects tend to review longer periods in the
low accuracy treatment. Although their actions partly depend on the information two
period ago in the high accuracy treatment, the dependencies are marginal in the aggregate
level.

6.5. Impact of Experiences
Several existing studies often reported that the frequency of cooperation changes as
people experience playing repeated games (see discussion in Dal Bó and Fréchette(2014)). Since the welfare is uniquely determined by the overall cooperation rates in our
payoff setting (the additive separability payoff structure), the shift of the overall
cooperation rates imply the shift of welfare of the two players.
To examine the impact of the experience of repeated games on overall cooperation
rates, we perform the reduced-form, linear regression analysis, explained in Appendix 2
in details. The results indicate that, although there are some experience effects on action
choices, the sizes of the effects are not remarkably large in our data. We also findqualitatively similar results on signal-contingent cooperation rates which are also reported
in Appendix 2.
We also investigate the effect of experience on retaliation intensities. Here we
perform a similar, reduced-form regression analysis, explained in Appendix 2 in details.

23

The results indicate that the retaliation intensities do not change remarkably as our
subjects gain experience.

7. Estimation of Individual Strategies - Methodology
In this and the following sections, we present the direct estimation of individual
strategies of our subjects. Given the recent consensus in the literature of experimental
repeated game that there is substantial heterogeneity of strategies subjects employ (seeDal Bó and Fréchette (2014) for a review), we do not specifically pick up only a single
class of strategy (i.e. g-TFT) and fit the data to determine the model parameters. Rather,
we list various strategies our subjects could take, and estimate the frequency that eachstrategy emerges among our subjects to assess the prevalence of the strategies. The
primary goal of the exercise here is to verify our findings in Section 6 and to perform
more detailed analyses on g-TFT strategies and retaliation intensities from the viewpointof individual strategies.
The methodology we employ here is the Strategy Frequency Estimation Method(SFEM) developed by Dal Bó and Fréchette (2011). The SFEM is a maximum likelihood
estimation (MLE) of a finite mixture model of strategies subjects individually use, wheremodel parameters to be estimated are the frequencies of each strategy emerging among
subjects, and the parameter J which controls stochastic mistakes of action choices orimplementation error, whose probability of occurrence is 1/ (1  exp(1/ J )) . The
details of the computation of the likelihood is provided in Appendix 3. The underlying
assumption is that each subject keeps employing a specific strategy across all repeatedgames in each treatment. The validity of this method is verified in the Monte-Carlosimulations in Fudenberg, Rand, and Dreber (2012), in which, closely related to our study,
they correctly dissociate g-TFT players from players employing other strategies.
In Appendix 4, we perform robustness checks of the SFEM estimates using only the
final two repeated games. The SFEM estimates have little changes in both treatments,
which would not occur if our subjects systematically changed their strategies acrossrepeated games. This is also consistent with our findings in Section 6 where we do notfind remarkable changes in either the cooperation rates or the retaliation intensities as our

24

subjects gain experiences across repeated games.
[TABLE 7 HERE]In the SFEM framework, the strategy set considered in the estimation are prespecified. Given the difficulty to cover all possible set of strategies, we only include thestrategies that share significant proportions in existing studies on experimental repeatedgames as well as g-TFT which is our primary focus. Table 7 displays the list of thestrategies in our SFEM. The list includes TFT, TF2T, TF3T, 2TFT, 2TF2T, Grim (trigger
strategy), 18 Grim-2, Grim-3, All-C (always cooperate) and All-D (always defect), which
are typically listed in the literature of infinitely repeated games in imperfect monitoring(Fudenberg, Rand, and Dreber (2012) for an instance). 19 Among them All-D is a noncooperative strategy, while other strategies (TFT, TF2T, TF3T, 2TFT, 2TF2T, Grim,
Grim-2, Grim-3, All-C) are regarded as cooperative strategies which play cooperation at
the starts of each repeated game, and keep cooperating unless they believe the opponent
might switch to defect. TF2T, TF3T, 2TF2T, Grim-2 and Grim-3 are “lenient” strategies(Fudenberg, Rand, and Dreber (2012)) that start punishing only after observing several
consecutive occurrence of bad signals. TF2T (TF3T) retaliates once after observing two(three) consecutive bad signals, and 2TF2T retaliate twice after observing two
consecutive bad signals, corresponding a simple form of so-called “review strategies”(lenient strategies with long-term punishments in the proof of the limit folk theorem (seeMatsushima (2004) and Sugaya (2012))). Grim-2 (Grim-3) is a lenient variant of Grim
strategy, which triggers to keep defecting after observing two (three) consecutive
deviations from (a, A), the combination of good signal from the opponent and own
cooperative choice.
Importantly, motivated with the theoretical importance of g-TFT in imperfect private
18

Here the definition of Grim strategy is modified to cover the private monitoring case where nocommon signals are observable. The player starts to keep playing defection if she observed a badsignal or she played defection in the previous round. Note that she could mistakenly play defection
before the “trigger” is pulled, since the implementation error of action choices are allowed in the
SFEM framework.
19 Literature often add D-TFT in the strategy set, which plays defection at round 1 then follows TFT
from Round 2. However, we do not find any significant frequency of D-TFT in both treatments in our
SFEM estimates even if we include it.

25monitoring, we also include g-TFT in the list. Inclusion of g-TFT is scant in the literatureof experimental repeated games. The only exception is Fudenberg, Rand, and Dreber(2012), in which they find that at least a certain share of their subjects follow g-TFT evenin imperfect public monitoring where g-TFT is relatively less important. However, their
discussion on g-TFT is merely incomplete since they include g-TFT just aiming toperform robustness checks for their main claims that are less relevant to g-TFT, and left
further discussions unanswered. Rather in this study, to address the implications of g-TFT
and associated retaliation intensities more rigorously in imperfect private monitoring, we
add many variants of g-TFT in our SFEM strategy to dissociate strategies with various
retaliation intensities.
Following Fudenberg, Rand, and Dreber (2012), we pre-specify probabilities ofretaliation, or equivalently, probabilities of cooperation after observing a bad signal in gTFT (i.e. r (b) ) in our strategy set. We allow the probabilities of cooperation given a badsignal to take nine distinct values with the increment by 12.5%, which are 100%, 87.5%,
75%, 62.5%, 50%, 37.5%, 25%, 12.5%, and 0%. Moreover, to cover the case that our
subjects might play defections even after observing a good signal (i.e. r (a )  1 ), we also
allow stochastic defections given a good signal. The probabilities of cooperation given agood signal (i.e. r (a ) ) are allowed to take the nine distinct values with the increment by
12.5%, which are 100%, 87.5%, 75%, 62.5%, 50%, 37.5%, 25%, 12.5% and 0%. Here gTFT- r (a ) - r (b) denotes the g-TFT which plays cooperation stochastically after
observing a good signal with the probability of r (a ) , and takes cooperation
stochastically after observing a bad signal with the probability of r (b) . 20 We list all thepossible combinations of r (a ) and r (b) in g-TFT in our strategy set as long as the gTFT has a non-negative retaliation intensity (i.e. r (a ) t r (b) ). Specifically we refer the gTFT strategies playing cooperation with constant probabilities r irrespective of signals
as Random strategies (equivalent to g-TFT- r - r ), denoted by Random- r , as primitive,
signal non-contingent, zero retaliation variants of g-TFT including All-C and All-D as the
special cases. We regard a g-TFT strategy as non-cooperative if the g-TFT is less20

For simplicity, we assume the probability of playing cooperation at round 1 coincides with the
choice probability given a good signal (i.e. r ( a ) ).

26

cooperative than Random-0.5. More specifically, a g-TFT is regarded as non-cooperativeif the r (a ) and r (b) in the g-TFT are no more than 0.5, otherwise the g-TFT strategies
are cooperative.
Besides including these g-TFT strategies in the list, we also add a family of g-2TFT
as the strategies which perform even stronger punishments than TFT. The motivation
comes from our earlier analysis in Section 6, in which we find that our subjects
aggregately take stronger retaliation intensities than the level implied by the standard
theory in the high accuracy treatment. The family of g-2TFT strategies (g-2TFT- r )allows the second retaliations to be stochastic (play cooperation with the probability ofr in the second punishment) as the generous variants of 2TFT. 21 Conservatively, we

also include a family of g-TF2T (g-TF2T- r ) as the generous variants of TF2T whichallow stochastic punishments if two consecutive bad signals occur (play cooperation withthe probability of r after observing two consecutive bad signals). 22

8. Estimation of Individual Strategies - Results
8.1. Cooperative Strategies and Non-Cooperative Strategies[Table 8 HERE][Table 9 HERE]
Table 8 presents the estimates for the frequencies of strategies our subjects follow,
and Table 9 displays the aggregated frequencies. First, we focus on the share ofcooperative strategies in both treatments. Our finding in Section 6 suggests that our21

Unlike g-TFT, in g-2TFT we do not allow defections until punishing phases start. Allowing
defections outside of punishing phases start reduce the retaliation intensities, which is contradictory
to the motivation for employing stronger (multi-round) punishments than the punishment attainable in
only one round. Also, the strategies are assumed to play cooperation at round 1 similarly to TFT, asjust multi-round punishing variants of TFT.
22 Unlike g-TFT, in g-TF2T we do not allow defections until punishing phases start, since defections
outside of punishing phases are contradictory to the motivation for employing lenient strategies thatallow “giving the benefit of the doubt” to an opponent on the first defection (Fudenberg et al. (2012)).
With the same reason, the strategies are assumed to play cooperation at round 1.

27

subjects take cooperative strategies in the high accuracy treatment, but do not necessarily
take cooperative strategies in the low accuracy treatment. Here we further examine thespecific shares of the cooperative strategies.
In the high accuracy treatment, cooperative strategies, i.e., other strategies than AllD, Random with the cooperation rate no more than 0.5 and g-TFT which are lesscooperative than Random-0.5 (g-TFT-0.5- r (b) , g-TFT-0.375- r (b) , g-TFT-0.25- r (b) ,and g-TFT-0.125- r (b) ), exceed non-cooperative strategies, i.e., All-D, Random with thecooperation rate no more than 0.5, and g-TFT with is less cooperative than Random-0.5(g-TFT-0.5- r (b) , g-TFT-0.375- r (b) , g-TFT-0.25- r (b) , and g-TFT-0.125- r (b) ). The
share of the cooperative strategies is 83.9%, and that of the non-cooperative strategies is
the rest, which is 16.1%. The latter share is statistically significantly smaller than theformer (p < 0.001). Although there are considerable heterogeneities in the strategies our
subjects follow as seen in Table 8, most of our subjects take cooperative strategies in thehigh accuracy treatment.
On the contrary, our subjects tend to take non-cooperative strategies more in the low
accuracy treatment. The share of cooperative strategies drops to 34.7% and the noncooperative strategies share 65.3%. The share of non-cooperative strategies in the low
accuracy treatment is statistically significantly larger than that in the high accuracy
treatment (p < 0.001). Individually, the share of All-D is the most, which is 19.0%.
The finding that the share of non-cooperative strategies is considerable in the low
accuracy treatment is consistent with the previous finding in Section 6 that the round 1
cooperation rate remains low, reaching to only 43.3% in the low accuracy treatment. Here,our SFEM estimates further indicate that indeed more than half of our subjects follow
non-cooperative strategies in the low accuracy treatment.
This result on the large share of non-cooperative strategies is somewhat surprising
given that the payoff parameters and discount factor are conductive to the g-TFT
cooperative equilibria even in the low accuracy treatment. The complexity of theequilibrium cooperative strategy is not the primary impediment to follow cooperative
strategies since the equilibrium cooperative strategy in the low accuracy treatment isapproximately TFT in our experimental setup, which is quite simple to implement. Rather,
our result implies that the poor signal quality in the low accuracy treatment somewhat

28strongly discourages our subjects to take cooperative strategies.
With respect to this point, Fudenberg, Rand, and Dreber (2012) reported in theirimperfect public monitoring experiment that the frequencies of cooperative strategies
drop significantly as the level of noise is increased from 1/16 to 1/8. Our study
demonstrates that, even by drastically changing signal noise from 1/10 to 4/10 inimperfect private monitoring, similarly the poorer signal quality indeed discourages our
subjects to take cooperative strategies even if the experimental parameters are conductive
to cooperate.
RESULT 1-b: Most of our subjects take cooperative strategies in the high accuracy
treatment, while in the low accuracy treatment, many of our subjects take non-cooperative
strategies.

8.2. Proportion of G-TFT Strategies
Secondly, motivated with the theoretical arguments in Section 4, we focus on theshare of g-TFT. Our SEFM estimates in Table 8 and Table 9 indicate that the family ofg-TFT shares a substantial proportion in our imperfect private monitoring. In the highaccuracy treatment, g-TFT-1-0.5 individually shares the most over the entire set ofstrategies, which is 17.1%, followed by another g-TFT strategy, which is g-TFT-0.8750.25 (10.7%). The total share of the family of g-TFT (g-TFT- r (a ) - r (b) including TFT,
but excluding signal non-contingent variants of g-TFT (i.e. All-C (g-TFT-1-1), All-D (gTFT-0-0) and Random- r (g-TFT- r - r ) ) is as large as 70.6%. The extended family of gTFT, which includes the signal non-contingent, primitive variants of g-TFT shares 76.8%.
Moreover, the further extend family of g-TFT, which includes the family of g-2TFT as
multi-round punishing variants, shares 77.7%. As shown in these large numbers, indeed
most of our subjects follow one of the strategies in the class of g-TFT.
The share of g-TFT is also substantially large in the low accuracy treatment. The
share of the family of g-TFT is 55.6%, and the extended family with signal noncontingent variants shares 94.6%, and the further extended family with multi-roundpunishing variants share 96.2%. Regardless of the treatment, we find the substantial share29

of our subjects following a strategy in the family of g-TFT.
This finding that the many subjects follow g-TFT regardless of treatments indicatethat their decisions on retaliation largely depends on a single occurrence of a bad signal(c.f. lenient strategies). This finding is consistent with our previous finding in Section 6that the action choices only marginally depend on the signals occurring two periods ago
in both treatments. 23
RESULT 4: Our SFEM estimates indicate that the family of g-TFT share substantial
proportion among the strategies our subjects follow in both treatments.

8.3. Retaliation Intensity
Thirdly, observing that many of our subjects follow g-TFT, we turn to the issue on
retaliation intensities which is one of our primary focuses of the study. Here, we address
how many fraction of our subjects take retaliation intensities which are consistent with gTFT equilibria. Recall that the level of retaliation intensity implied by the g-TFT
equilibria is 0.235 in the high accuracy treatment. Approximately, g-TFT-1-0.75, g-TFT0.875-0.625, g-TFT-0.75-0.50, g-TFT-0.625-0.37.5, g-TFT-0.50-0.25, g-TFT-0.3750.12.5, and g-TFT-0.25-0 have this intensity of retaliation in our strategy set. However,
our SFEM estimates in Table 8 and Table 9 indicate that the joint share of these strategies
is only 5.6% which does not significantly differ from zero (p = 0.319), implying that very
few of our subjects follow equilibrium g-TFT strategies in the high accuracy treatment.
This finding that only a limited number of our subjects follow equilibrium retaliation
intensity is also true in the low accuracy treatment. In the low accuracy treatment, theretaliation intensity implied by the g-TFT equilibria is much larger than that in the high23
The result that our SFEM estimates find many of our subjects playing g-TFT in our imperfectprivate monitoring is seemingly less consistent with the finding of Fudenberg, Rand, and Dreber(2012), which found only a small proportion of their subjects playing g-TFT in their imperfect public
monitoring. However, we are not able to address the exact factors behind the discrepancy between our
results and theirs, since our experimental settings are different from theirs in many aspects such as
payoff parameters, discount factor, signal accuracy. Perhaps most importantly our setting is a private
monitoring environment in which g-TFT play important roles as a benchmark equilibrium strategy.
Nonetheless as we discuss later in this section, similarly to Fudenberg, Rand, and Dreber (2012), we
find a certain proportion of our subjects following lenient strategies in the high accuracy treatment.

30

accuracy treatment, which is 0.94. Approximately, the retaliation intensity in TFT isequivalent to the level implied by the g-TFT equilibria. However, the share of TFT in the
low accuracy treatment is only 2.7%, again, which is not significantly different from zero(p = 0.221). These results demonstrate that, despite that many of our subjects follow oneof the g-TFT strategies, almost all of them do not follow the retaliation intensities impliedin the g-TFT equilibria in both treatments.
Observing that almost all of our subjects do not follow the retaliation intensities
implied by the g-TFT equilibria, we further address how they tend to deviate from thetheory. In Section 6 we find that the aggregate level of retaliation intensity in the highaccuracy treatment is larger than that implied by the g-TFT equilibria. Now we examine
whether the similar, consistent results emerge here in the SFEM estimates.
Our SFEM estimates in Table 8 and Table 9 indicate that the group of stronger
retaliation variants of g-TFT (g-TFT strategies with retaliation intensities more than 0.25
i.e. g-TFT-1-0.625/0.5/0.375/0.25/0.125/0, g-TFT-0.875-0.5/0.375/0.25/0.125/0, g-TFT0.75-0.375/0.25/0.125/0, g-TFT-0.625-0.25/0.125/0, g-TFT-0.5-0.125/0, g-TFT-0.375-0
and g-2TFT-0.875/0.75/0.625/0.5/0.375/0.25/0.125/0) jointly shares 54.4% in the highaccuracy treatment. Previously we find slightly more than 70% of our subjects follow gTFT. However, the results here imply that roughly three fourth of them retaliate strongly
more than the g-TFT equilibria require. Indeed, the share of the weaker retaliation variantsof g-TFT, even including signal non-contingent strategies, merely reaches to 17.7% (gTFT-1-0.875, g-TFT-0.875-0.75, g-TFT-0.75-0.625, g-TFT-0.625-0.5, g-TFT-0.5-0.375,
g-TFT-0.375-0.25, g-TFT-0.25-0.125, g-TFT-0.125-0 and signal non-contingent, zero
retaliation strategies, which are All-C, All-D, and Random- r ), which is statisticallysignificantly smaller than the share by the group of stronger retaliation variants (p <0.001).
On the other hand in the low accuracy treatment, we previously find weaker
retaliation intensities in the aggregate level in Section 6. Now we examine how many of
our subjects follow the weaker retaliation strategies in our SFEM estimates. Our SFEM
estimates indicate that the group of weaker retaliation variants of g-TFT (g-TFT otherthan TFT and signal non-contingent, zero retaliation variants of g-TFT which are All-C,
All-D, and Random- r ) jointly shares 93.5%. Again, we have found that 96.2% of our
subjects take strategies in g-TFT, however, weaker retaliation variants of g-TFT

31

predominantly share almost all of them. On the contrary, strong retaliation variants of gTFT (here the multi-round punishing variants of TFT, which are g-2TFT0.875/0.75/0.625/0.5/0.375/0.25/0.125/0) share less than 0.1%, which is statisticallysignificantly smaller than the share by the weaker retaliation variants (p < 0.001).
Previously in Section 6 we found that the mean retaliation intensities deviate fromthe values implied by the g-TFT equilibria in the aggregate level systematically. Here we
further examine whether the finding still hold even if we restrict our attention to the
behavior of g-TFT players rather than the aggregate behavior with the entire strategies.
To focus on the behavior of the g-TFT players, we compute the mean retaliation
intensities conditional on the g-TFT strategies (including All-C, All-D, and Random- r ).
The conditional mean retaliation intensity in the high accuracy treatment is 0.426 (s.d.
0.033), again which is significantly larger than the value predicted by the g-TFT equilibria(0.235, p < 0.001). In the low accuracy treatment the mean retaliation intensity is 0.148(s.d. 0.025), again which is significantly smaller than the value implied by the g-TFT
equilibria (0.94, p < 0.001). The direct comparison of the two mean retaliation intensities
demonstrate that the conditional mean of retaliation intensities in the high accuracy
treatment is significantly larger than that in the low accuracy treatment (p < 0.001). Even
restricted to the behavior of g-TFT players, still their behavior deviate from the theoreticalpredictions systematically similarly to the findings in Section 6.
RESULT 2-b: Our SFEM estimates indicate that, in both treatments, only a small numberof our subjects follow the retaliation intensities implied by the g-TFT equilibria. In thehigh accuracy treatment, the share of stronger retaliation variants of g-TFT outweigh thatof weaker retaliation variants. In the low accuracy treatment, the share of weaker
retaliation variants of g-TFT outweigh that of stronger retaliation variants. Moreover, themean retaliation intensity among g-TFT players is larger in the high accuracy treatment
than the value implied by the g-TFT equilibria, and is smaller in the low accuracy
treatment than that by the equilibria. Furthermore, the mean retaliation intensity by the gTFT strategies is larger in the high accuracy treatment than that in the low accuracy
treatment opposed to the theoretical implications by the g-TFT equilibria.

32

8.4. Long-Term Punishment
Fourthly, we focus on long-term punishing strategies. We have observed that theaggregate level of retaliation intensity is smaller than the level implied by the g-TFT
equilibria in the low accuracy treatment in Section 6. Here we address the concern thatthe seemingly weak retaliation intensity might spuriously arise when some subjects
employ long-term punishing strategies.
Our SEFM estimates in Table 8 and Table 9 indicate this is not the case in our data.
The share of the strategies with long-term punishments (a family of 2TFT, 2TF2T, Grim,
Grim2 and Grim 3) in the low accuracy treatment is less than 3% which is statisticallyinsignificant (p = 0.435), hence the effect of strategies with long-term punishment isminimum. This is also true in the high accuracy case, where the joint share by the longterm punishing strategies is 6.6%, again, which is statistically insignificant (p = 0.203)
RESULT 2-c: The share of strategies with long-term punishments, which could
spuriously reduce the observed retaliation intensities, is small in both treatments.

8.5. Lenience
Finally, we focus on the share of lenient strategies. As seen in Section 6, our subjects
do not have tendency to rely on longer histories in the low accuracy treatment. Thisfinding suggests that the share of lenient strategies might not be larger in the poorer
monitoring technologies in our imperfect private monitoring, opposed to the speculationsin the field of review strategies (Radner (1986), Matsushima (2004), and Sugaya (2012)).
Here we examine whether the share of lenient strategies becomes larger as the monitoring
technology becomes poorer.
Our SFEM estimates in Table 8 and Table 9 indicate that, in the low accuracy
treatment, none of the individual shares by the lenient (review) strategies (the family of
g-TF2T including TF2T, TF3T, 2TF2T, Grim-2 and Grim-3) significantly differ fromzero. Even jointly, they share only 3.8%, which is not significantly different from zero (p= 0.359). The lenient strategies do not share a remarkable proportion in the low accuracy

33

treatment. On the contrary, in the high accuracy treatment, the share of lenient strategies
raises to 21.4% which is significantly different from zero (p = 0.023) and is marginallysignificantly larger than that in the low accuracy treatment (p = 0.055). Thus, we conclude
that there is no tendency to have more lenient strategies in the poorer monitoring
technologies.
RESULTS 3-b:

Our SFEM estimates provide no evidence for a larger share of lenient

strategies in the low accuracy treatment. Rather, the share is only negligible in the low
accuracy treatment, while they share approximately 20% in the high accuracy treatment.

9. Feedback to Theory
Our experimental results indicate that the cooperation rate is greater in the highaccuracy treatment, a substantial proportion of the subjects play g-TFT strategies, and theretaliation intensity is greater in the high accuracy treatment.
As a feedback from these experimental findings to the theoretical development, wedemonstrate an alternative theory, which is more consistent with observed behavior than
the standard theory. This section ignores the heterogeneity of strategies the subjects
employed. We replace this heterogeneity with the common knowledge assumption on thestrategy that players employ.
The purpose of this section is to associate our subjects’ behavior with their aspectsof psychology and bounded rationality. To be more precise, we permit each player to be
motivated by not only pure self-interest but also reciprocity. We permit each player to beoften naïve enough to select actions at random. We further permit the degrees of such
reciprocity and naiveté to be dependent on the level of monitoring accuracy.
By incorporating reciprocity and naïveté into equilibrium analysis, we characterize
the underlying behavioral model of preferences that makes the retaliation intensity
implied by the g-TFT equilibria increasing in the level of monitoring accuracy, i.e., moreconsistent with our experimental findings.
In order to focus on the incentives at the second and later rounds, this section will
simply write (r (a ), r (b)) instead of (q, r (a ), r (b)) for any g-TFT strategy. Fix an34arbitrary p  ( 1 ,1) as the lower bound of monitoring accuracy.
2

9.1. Behavioral Model
Consider an arbitrary accuracy-contingent g-TFT strategy, which is denoted by(r (a; p ), r (b; p)) p( p ,1) .
For each level of monitoring accuracy p  ( p,1) , a player makes stochastic action
choices according to the g-TFT strategy (r (a ), r (b)) (r (a; p ), r (b; p )) . We assume thatthe player selects both action A and action B with positive probabilities, i.e.,(6)0  r (a; p )  1 and 0  r (b; p )  1 .

We introduce naïveté as follows. At every round, with a probability 2H ( p )  [0, 1 2 ] ,the player naively, or randomly, selects between action A and action B, i.e.,min[r (a; p ),1  r (a; p ), r (b; p ),1  r (b; p )] t H ( p ) .

With the remaining probability 1  2H ( p ) , the player makes the action choice in the more
conscious manner. 24 25
We introduce reciprocity as follows. Suppose that the player observes signal a , i.e.,
the good signal for his opponent. He feels guilty when he selects the defective action
despite the observation of the good signal. In this case, he can save the psychological costc(a; p ) t 0 by selecting the cooperative action. Hence, the instantaneous gain fromselecting action B is equal to Y  c(a; p ) , while the resultant future loss is equal to

G Z (2 p  1){r (a; p )  r (b; p )} . From (6), we require the following properties as a part of24

In order to calm the tense relation between rationality and empirical data, economic theory andempirics have used stochastic choice models such as logit and probit models that incorporated random
error into equilibrium analysis. For instance, the model of quantal response equilibrium assumed thatthe deviation errors from the optimal action choice are negatively correlated with the resultant payoffs.
See Goerree, Holt, and Palfrey (2008) for a review. In contract, this section assumes that the deviation
errors induced by naïveté is independent of either the resultant payoff or the observed signal but
depends on the level of monitoring accuracy.
25 The general experimental literature has often pointed out that social preference facilitates
cooperation. See Güth, Schmittberger, and Schwarze (1982), Berg, Dickhaut, and McCabe (1995), andFehr and Gächter (2000). The literature assumes that preferences depend on various contexts of thegame being played. See Rabin (1993), Charness and Rabin (2002), Falk, Fehr, and Fischbacher (2003),
Dufwenberg and Kirchsteiger (2004), and Falk and Fishbacher (2005). This paper makes the relevant
context parameterized by the level of monitoring accuracy.

35

the equilibrium constraints in behavioral theory:(7)[ Y  c(a; p ) ! G Z (2 p  1){r (a; p )  r (b; p )} ]э[ r (a; p ) H ( p ) ],

and
(8)

[ Y  c(a; p )  G Z (2 p  1){r (a; p )  r (b; p )} ]э[ r (a; p) 1  H ( p) ]. 26
Next, suppose that the player observes signal b , i.e., the bad signal for his opponent.

He is in an ill-temper when he selects the cooperative action despite the observation of
the bad signal. In this case, he can save the psychological cost c(b; p ) t 0 by selecting
the defective action. Hence, the instantaneous gain from selecting action B is equal toY  c(b; p ) , while the resultant future loss is equal to G Z (2 p  1){r (a; p )  r (b; p )} .

From (6), we require the following properties as the other part of this section’s
equilibrium constraints:(9)[ Y  c(b; p ) ! G Z (2 p  1){r (a; p )  r (b; p )} ]э[ r (b; p ) H ( p ) ],

and
(10)

[ Y  c(b; p )  G Z (2 p  1){r (a; p )  r (b; p )} ]э[ r (b; p) 1  H ( p) ].
We define a behavioral model as(H ( p), c(a, p), c(b; p)) p( p ,1) .
An accuracy-contingent g-TFT strategy (r (a; p ), r (b; p )) p( p ,1)

is said to be an

equilibrium in the behavioral model (H ( p ), c(a, p), c(b; p)) p( p ,1) if the properties (7), (8),(9), and (10) hold for all p  ( p,1) . 27

9.2. Characterization
A player with behavioral model (H ( p ), c(a, p), c(b; p)) p( p ,1) is said to be positively
reciprocal in accuracy p if c(a; p ) ! 0 . She is said to be negatively reciprocal in p

Without any substantial change, we can interpret c(a; p ) as the psychological benefit, where hefeels better by selecting the cooperative action instead of the defective action after observing the goodsignal. We have the same point about c(b; p ) .
27 For simplicity of arguments, we eliminate the incentive constraints for the first-round action choices,and then focus on the action choices at the second and later rounds.
26

36

if c(b; p ) ! 0 . She is said to be null-reciprocal in p if c(a; p ) 0 and c(b; p )0.

We assume that c(a; p ) and c(b; p ) are continuous in accuracy p , andeither c(a; p ) 0 or c(b; p )0.

With this assumption, we can divide players into three categories, i.e., positively
reciprocal players, negatively reciprocal players, and null-reciprocal players.
According to our experimental findings, we assume that(i)both r (a; p ) and r (b; p ) are increasing and continuous in p ,

and
(ii)

the retaliation intensity r (a; p )  r (b; p ) is increasing in p .

Since w( p ) is decreasing in p , it follows from (ii) that there exists a critical levelpˆ  [ p,1] such thatr (a; p )  r (b; p ) ! w( p )if p ! pˆ ,r (a; p )  r (b; p )  w( p )if p  pˆ .

andThe retaliation intensity is greater than the retaliation intensity implied by the g-TFT
equilibria in standard theory if the level of monitoring accuracy is better than the criticallevel p̂ , while the retaliation intensity is lesser than the retaliation intensity implied by
the g-TFT equilibria in standard theory if the level of monitoring accuracy is worse thanthe critical level p̂ .
The following theorem shows that the above-mentioned equilibrium constraints, i.e.,(7), (8), (9), and (10), uniquely determine the underlying behavioral model.
Theorem 2: The accuracy-contingent g-TFT strategy (r (a; p ), r (b; p )) p( p ,1) is anequilibrium in the behavioral model (H ( p ), c(a, p), c(b; p)) p( p ,1) if and only if(11)

r (a; p ) 1  H ( p ) and r (b; p ) 1  H ( p )  w( p ) for all p ! pˆ ,andc(b; p )G (2 p  1) Z37r (a; p) H ( p )  w( p ) (12)Proof:c(a; p)and r (b; p ) H ( p ) for all p  pˆ .
G (2 p  1) Z

See Appendix 5.

From Theorem 2, the behavioral model (H ( p ), c(a, p), c(b; p)) p( p ,1) is uniquelydetermined: for every p ! pˆH ( p ) 1  r ( a; p ) ,c(a; p )0 , andc(b; p ) G (2 p  1) Z {r (a; p )  r (b; p )  w( p )} ,and for every p  pˆ ,H ( p ) r (b; p) ,c(a; p ) G (2 p  1) Z {w( p )  r (a; p )  r (b; p )} , andc(b; p )0.

Note that(iii)

the player is null-reciprocal at the critical level p̂ , i.e., c(a; pˆ ) c(b; pˆ ) 0 ,(iv)

H ( p ) is single-peaked with the peak at the critical level p̂ , i.e., increasing inp  ( p, pˆ ) and decreasing in p  ( pˆ ,1) ,(v)

c(a; p ) is decreasing in p  ( p, pˆ ) ,

and
(vi)

c(b; p ) is increasing in p  ( pˆ ,1) .

9.3. Trade-Offs
The behavioral model (H ( p ), c(a, p), c(b; p)) p( p ,1) is said to be more kind in p
than in pc if either
it is more positively reciprocal in p than in pc , i.e., c(a; p ) ! c(a; pc) ,or
it is less negatively reciprocal in p than in pc , i.e., c(b; pc) ! c(b; p ) .

38

From Theorem 2, the behavioral model has the following trade-off between kindness andaccuracy:(vii)

The less kind a player is, the more accurate the monitoring technology is.

Given a sufficient level of monitoring accuracy, players tend to be more negatively
reciprocal as monitoring is more accurate. This tendency makes the retaliation intensity
severer, and therefore works against the better success in cooperation caused by the
improvement of monitoring technology. Given an insufficient level of monitoring
accuracy, they tend to be more positively reciprocal as monitoring is less accurate. Thistendency makes the retaliation intensity milder, and therefore mitigates the worse successin cooperation caused by the deterioration of monitoring technology. 28
From Theorem 2, the behavioral model also has the following trade-off betweennaïveté and reciprocity:(viii)The more naively a player makes action choices, the less reciprocal he is.

Given that a player is negatively reciprocal, he tends to be more conscious, i.e., less likely
to mistakenly select the defective action despite observing the good signal, as he is morenegatively reciprocal. Given that a player is positively reciprocal, he tends to be more
conscious, i.e., less likely to mistakenly select the cooperative action despite observing
the bad signal, as he is more positively reciprocal.

9.4. Uniqueness of G-TFT Equilibrium
We further show that the accuracy-contingent g-TFT strategy (r (a; p ), r (b; p )) p( p ,1)
is the only plausible accuracy-contingent g-TFT equilibrium in the derived behavioral
model (H ( p ), c(a, p), c(b; p)) p( p ,1) . In fact, whenever p  pˆ , then there exists no other28
The model here and that in Duffy and Muñoz-García (2012) share the common spirit that the social
preferences play an important role in people making collusion in repeated games. However, theimplications of the social preferences are different. Duffy and Muñoz-García demonstrated that,
motivated with fairness, social preference facilitates people to achieve collusion even if the time
discount factor is insufficient. In contrast, our behavioral theory shows that it crucially depends on themonitoring technology whether the social preference helps collusion or not due to the trade-off
between the kindness and accuracy; motivated with reciprocity, the social preference facilitate people
to collude even under poorer monitoring accuracies, where a larger discount factor is required.
However, it inhibits people to collude under better monitoring accuracies.

39

accuracy-contingent g-TFT equilibrium strategy at all.
If ppˆ , there exists another accuracy-contingent g-TFT strategy equilibrium,(r (a; p ), r (b; p )) p( p ,1) , which is specified asr (a; p ) H ( p )  w( p ) and r (b; p ) H ( p ) for all p  ( p,1) ,where (r (a; p ), r (b; p )) p( p ,1) and(r (a; p ), r (b; p)) p( p ,1) are the only equilibria in thederived behavioral model.
Note that (r (a; p ), r (b; p )) p( p ,1)

is quite implausible, because r (a; p ) andr (b; p ) are decreasing, and the frequency of cooperative action choices is therefore

decreasing, in level of monitoring accuracy. Note also that (r (a; p ), r (b; p )) p( p ,1) is lessefficient than (r (a; p ), r (b; p )) p( p ,1) , irrespective of p  ( p,1) .
The above-mentioned uniqueness is in contract with the standard theory, which has
numerous g-TFT equilibria. The uniqueness substantially relies on the inclusions of thepsychological costs c(a; p ) and c(b; p ) , which make the equilibrium constraints
dependent on the signal occurrence at the previous round.

10. Conclusion
This paper experimentally examines collusion in repeated prisoners’ dilemma withrandom termination where monitoring is imperfect and private; each player obtains
information on the opponent’s action choice through a signal instead of a direct
observation, and the signal the opponent observes is not observable to the player. We
assume that the continuation probability is large enough to support collusion even in thepoor monitoring technology. Our study is the first experimental attempt to investigate
imperfect private monitoring.
Our experimental results indicate that a significant proportion of our subjects
employed g-TFT strategies, which are the straightforward stochastic extensions of thewell-known TFT strategy. We significantly depart from the experimental literature byfocusing on g-TFT strategies, which have attracted less attention in the empiricalliterature despite its theoretical importance. Our findings that a significant proportion of

40

our subjects follow g-TFT strategies reveal its empirical importance.
Although many subjects follow g-TFT strategies, their retaliating policies
systematically deviate from the predictions by the g-TFT equilibria. Our subjects retaliate
more in the high accuracy treatment than in the low accuracy treatment, contrarily to thetheoretical implications. They retaliate more in the high accuracy treatment, while they
retaliate lesser in the low accuracy treatment. These experimental findings indicate thatthe subjects fail to improve their welfare by effectively utilizing the monitoring
technology as the standard theory predicted.
As a feedback from these experimental findings to the theoretical development, wecharacterize a behavioral model of preferences that incorporates reciprocity and naïveté.
Our behavioral theory describes the signal-contingent behavior consistent with ourexperimental results as the unique, plausible g-TFT equilibrium. This feedback could be
expected as a clue to establish a pervasive theory that has more relevancies to real
behavior and more predictive power.

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43

Table 1:
Prisoner’s Dilemma with Symmetry and Additive Separability( X , Y , Z ) (60,10,55)
A
A
B

B
5 70
15 15

60 60
70 5

Table 2:Features of Experimental Design
October 5, 2006 (10:30 ~ 12:30)
October 5, 2006 (14:30 ~ 16:30)
October 6, 2006 (10:30 ~ 12:30)
October 6, 2006 (14:30 ~ 16:30)

N
Cooperative-choice
Good signal

8,864
8,864

Number of Subjects
28
24
28
28

Treatment order (sequence of game lengths)
0.6 (24, 40, 25), 0.9 (28, 33, 14)
0.6 (20, 23, 37), 0.9 (34, 34, 19)
0.9 (38, 21, 25), 0.6 (25, 28, 29)
0.9 (25, 35, 23), 0.6 (36, 30, 21)

Table 3:
Decisions and Signal
p 0.9
Mean
St.
Dev.
0.672
0.637

0.469
0.481

N

p 0.6
Mean St. Dev.

9,144
9,144

0.355
0.483

Table 4:Means of Cooperative Action Choice
p 0.9
p 0.6
q( p ) (round 1)r ( a; p )Individual-level meansr (b; p )Individual-level means

0.479
0.500

p-values

0.781 (0.042)

0.438 (0.035)< 0.001

0.852 (0.033)
0.799 (0.032)
0.344 (0.026)
0.448 (0.026)

0.437 (0.018)
0.423 (0.027)
0.272 (0.030)
0.279 (0.032)<0.001< 0.001+< 0.001< 0.001+

Notes: The standard errors (shown in parentheses) are block-bootstrapped (subject and repeatedgame level) with 5,000 repetitions, which is used to calculate p-values. The null hypothesis is thatthe values are identical across the two treatments.
+ Furthermore, Wilcoxon matched-pair test within individual rejects the null hypothesis that thevalues are identical across the two treatments (p < 0.001 for each).

44

Table 5:Retaliation Intensities
Mean
0.508

S.E.
0.028

p-value< 0.001+

Individual-level means

0.352
0.165

0.028
0.025

< 0.001+< 0.001+

Individual-level means(r (a; 0.9)  r (b; 0.9))  (r (a; 0.6)  r (b; 0.6))

0.144
0.344

0.023
0.031

< 0.001+< 0.001

Individual-level means

0.208

0.029

< 0.001r (a; 0.9)  r (b; 0.9)
r (a; 0.6)  r (b; 0.6)

Notes: The standard errors are block-bootstrapped (subject and repeated game level) with 5,000
repetitions, which is used to calculate p-values.
+ Hypothesis tests for the comparison to the value implied by the standard theory (w(p)), which
is 0.235 in the high accuracy treatment and 0.94 in the low accuracy treatment. The null
hypothesis is that the mean is identical to the implied value.

Table 6:
Reduced form regression results on signal contingency
aA
aB
bA
bB
aAa
aBa
bAa
bBa

p = 0.9
0.887*** (0.019)
0.502*** (0.040)
0.506*** (0.038)
0.106*** (0.021)
0.062*** (0.022)
0.017 (0.054)
0.125*** (0.041)
0.132*** (0.037)

p = 0.6
0.766*** (0.038)
0.255*** (0.035)
0.582*** (0.049)
0.110*** (0.042)
0.037 (0.031)-0.055 (0.034)-0.045 (0.046)
0.022 (0.016)

Observations
R2
Adjusted R2

8,216
0.816
0.816

8,496
0.531
0.531

Notes: xYz in the regressors denotes that the player takes action Y, observes signal x on the
opponent’s choice, and the player has observed signal z in the previous round. Similarly, xY
denotes that the player takes action Y and observes signal x on the opponent’s choice. The
standard errors (shown in parentheses) are block-bootstrapped (subject and repeated game level)with 5,000 repetitions, which is used to calculate p-values. *p < 0.1, **p < 0.05, ***p < 0.01.

45

Table 7:
Strategy set in our SFEM
Strategy
All-C
TFT
g-TFT- r ( a ) - r (b)
All-D
TF2T
g-TF2T- r
TF3T
2TFT
g-2TFT- r
2TF2T
Grim
Grim-2
Grim-3
Random- r

Description
Always cooperate
Tit-For-Tat
Generous Tit-For-Tat, cooperate if a good signal occurs with the probability of r (a ) ,and forgive a bad signal and cooperate with the probability of r (b)Always defectTit for Two Tat (retaliate if bad signals occur in all of the last two rounds)
Generous Tit for Two Tat, playing cooperation stochastically with probability of reven after observing two consecutive bad signals
Tit for Three Tat (retaliate if a bad signal occurs in all of the last three rounds)
Two Tit-For-Tat (retaliate twice consecutively if a bad signal occurs)
Generous-Two Tit-For-Tat, retaliate for sure if a bad signal occurs, however, forgive
and cooperate with the probability of r in the next round if a good signal occurs(second punishment)
Two Tit for Two Tat (retaliate twice consecutively if a bad signal occurs in all of thelast two rounds)
Cooperate until player chose defection or observed a bad signal, then play defection
forever
Cooperate until the case happens twice in a row in which player chose defection or
observed a bad signal, then play defection forever
Cooperate until the case happens three times in a row in which player chose defection
or observed a bad signal, then play defection forever
Cooperate with the probability of r irrespective of signals

46

Table 8:Maximum Likelihood Estimates of individual strategies
All-C (g-TFT-1-1)
g-TFT-1-0.875
g-TFT-1-0.75
g-TFT-1-0.625
g-TFT-1-0.5
g-TFT-1-0.375
g-TFT-1-0.25
g-TFT-1-0.125
TFT (g-TFT-1-0)
Ramdom-0.875 (g-TFT-0.875-0.875)
g-TFT-0.875-0.75
g-TFT-0.875-0.625
g-TFT-0.875-0.5
g-TFT-0.875-0.375
g-TFT-0.875-0.25
g-TFT-0.875-0.125
g-TFT-0.875-0
Random-0.75 (g-TFT-0.75-0.75)
g-TFT-0.75-0.625
g-TFT-0.75-0.5
g-TFT-0.75-0.375
g-TFT-0.75-0.25
g-TFT-0.75-0.125
g-TFT-0.75-0
Random-0.625 (g-TFT-0.625-0.625)
g-TFT-0.625-0.5
g-TFT-0.625-0.375
g-TFT-0.625-0.25
g-TFT-0.625-0.125
g-TFT-0.625-0
Random-0.5 (g-TFT-0.5-0.5)
g-TFT-0.5-0.375
g-TFT-0.5-0.25
g-TFT-0.5-0.125
g-TFT-0.5-0
Random-0.375 (g-TFT-0.375-0.375)
g-TFT-0.375-0.25
g-TFT-0.375-0.125
g-TFT-0.375-0
Random-0.25 (g-TFT-0.25-0.25)
g-TFT-0.25-0.125
g-TFT-0.25-0
Random-0.125 (g-TFT-0.125-0.125)
g-TFT-0.125-0
All-D (g-TFT-0-0)
g-TF2T-0.875
g-TF2T-0.75
g-TF2T-0.625
g-TF2T-0.50
g-TF2T-0.375
g-TF2T-0.25
g-TF2T-0.125

p = 0.900000.171***
0
0.041
0
0.026
0.002
0.033
0.022
0.042
0.055
0.107***
0
0
0
0
0
0.003
0
0.011
0.009
0.011
0
0.023
0
0.026
0
0
0.053
0.003
0
0.019
0.021
0
0
0
0
0
0.008
0
0.029
0.028
0.074*
0
0
0.022
0
0
0

S.E.
(0.029)(0.009)(0)(0.040)(0.063)(0)(0.031)(0.007)(0.016)(0.014)(0.024)(0.033)(0.041)(0.048)(0.041)(0.006)(0)(0)(0.001)(0.017)(0.023)(0)(0.015)(0.008)(0.010)(0.005)(0.029)(0)(0.021)(0.010)(0.013)(0.034)(0.011)(0)(0.014)(0.019)(0.006)(0)(0.003)(0)(0)(0.014)(0.001)(0.021)(0.017)(0.040)(0.021)(0.010)(0.028)(0.020)(0.024)(0.001)

p = 0.6
0.037**
0
0
0
0
0
0
0.010
0.027
0
0.027
0
0.004
0
0
0.010
0.010
0
0.038
0.017
0.026
0
0
0
0.014
0
0.089*
0
0
0
0.023
0.050
0.038
0
0
0.044
0.090
0
0.014
0.063
0.069*
0.001
0.033
0.037
0.190***
0
0
0
0
0
0.009
0

S.E.
(0.019)(0)(0.002)
(0)
(0)
(0)
(0)
(0.014)(0.022)(0)(0.017)(0.015)(0.015)(0)(0)(0.011)(0.009)(0)(0.024)(0.022)(0.027)(0.004)(0)(0)(0.018)(0.012)(0.046)(0.015)(0)(0)(0.024)(0.045)(0.029)(0.006)(0)(0.045)(0.060)(0.012)(0.013)(0.053)(0.041)(0.011)(0.028)(0.022)(0.035)(0)(0)(0.003)(0)(0)(0.009)(0)

47

TF2T (g-TF2T-0)
TF3T
g-2TFT-0.875
g-2TFT-0.75
g-2TFT-0.625
g-2TFT-0.50
g-2TFT-0.375
g-2TFT-0.25
g-2TFT-0.125
2TFT (g-2TFT-0)
2TF2T
Grim
Grim-2
Grim-3
Gamma

0
0.070
0
0.009
0
0
0
0
0
0
0
0.009
0.013
0.035
0.280***(0.004)(0.045)(0)(0.016)(0.024)(0.005)
(0)
(0)
(0)
(0)
(0.013)(0.008)(0.016)(0.025)(0.035)

0
0
0
0
0
0
0
0
0
0
0.009
0
0
0.020
0.247***(0)(0.004)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0.010)(0)(0.007)(0.024)(0.059)Notes: The standard errors (shown in parentheses) are cluster-bootstrapped (individual-level) with
100 repetitions, which is used to calculate p-values. *p < 0.1, **p < 0.05, ***p < 0.01.

Table 9:Frequency of Cooperative strategies, g-TFT, Retaliation Intensities (RI)
Cooperative strategies
Non-cooperative strategies
Family of g-TFT
Including signal non-contingent, zero
RI strategies (All-C, All-D, Randomr)
Including multi-round punishingstrategies (g-2TFT)

Equilibrium RI
Stronger RI
Weaker RI (positive RI only)
Including zero RI strategies(RI = 0.25)(RI > 0.25)(0 < RI < 0.25)(RI < 0.25)

p = 0.9
83.9%
16.1%

p = 0.6
34.7%
65.3%

70.6%

55.6%

76.8%

96.2%

77.7%

96.2%

5.6%
54.4%
11.4%
17.7%

(RI = 1)
(RI > 1)(0 < RI < 1)
(RI < 1)

2.7%
0.0%
53.0%
93.5%

Mean RI conditional on g-TFT
Median RI conditional on g-TFT

0.426
0.5

0.148
0.125

Lenient strategies

21.4%

3.7%

48

For Online Publication
Appendix 1: Screens in Constant Random Termination

Figure A. 1: Screen when the repeated game continues

Figure A. 2: Screen when the repeated game is terminated49

Appendix 2: Impact of Experience
Several existing studies often reported that the frequency of cooperation changes as
people experience playing repeated games (see discussion in Dal Bó and Fréchette(2014)). The overall cooperation rates might rise with experiences of repeated games
when the experimental parameters are conductive for cooperation. Since the welfare isuniquely determined by the overall cooperation rates in our payoff setting (i.e. the additive
separability payoff structure), the shift of the overall cooperation rates imply the shift ofwelfare of the two players.
To examine the impact of the experience of repeated games on overall cooperation
rates, we perform the following reduced-form, linear regression analysis; we regress theaction choices on the two explanatory variables, “RG2” and “RG3”. The dummy variable“RG2” (“RG3”) takes one if the choice is made in the second (third) repeated games ineach treatment. Moreover, to further examine the change of the cooperation rate with-ina repeated game, we also include the dummy variable “First 14 Rd.” which takes one if
the choice is made in the first 14 rounds of each repeated game. The regression model isa fixed-effect model in which the individual heterogeneity in tendencies to takecooperative choices is controlled by individual fixed effects.
[TABLE A. 1 HERE]
Table A. 1 displays the regression results. In the high accuracy treatment, thecoefficient on the second repeated game is 0.029 (insignificant), and that on the third
repeated game is 0.105 (significant, p < 0.001). These positive values in the regression
coefficients indicate that our subjects tend to cooperate more as they gain experience,indicating that the welfare of the two players is improved by experience. However thesize is at most 11%, which is only a marginal effect.
In the low accuracy treatment, the coefficient on the second repeated game is -0.071(significant, p < 0.05), and that on the third repeated game is -0.118 (significant, p <0.001). On the contrary to the case in high accuracy treatment, our subjects tend to
become less cooperative as they gain experience even in the situation where the50experimental parameters are conductive for cooperation, rather indicating that the welfare
is worsen by the experience. However, the sizes of the effect of experience is at most 12%,
again, which is only a marginal effect.
As to the effect with-in a repeated game, the coefficients on the first 14 rounds are
statistically significantly larger than zero in both treatments (p < 0.001 for both
treatments), although the sizes are at most 8%. Our subjects tend to become lesscooperative as the rounds proceed in each repeated game, however the effect is small.
These results indicate that, although there are some experience effects on action
choices, however, the sizes of the effects are not remarkably large in our data. We also
perform the identical analysis to the signal contingent cooperation rates, and findqualitatively similar results (Table A. 2 and Table A. 3).
[TABLE A. 2 HERE][TABLE A. 3 HERE]
In addition to the cooperation rate, we also investigate the effect of experience on
retaliation intensities. Here we perform a similar, reduced-form regression analysis,
regressing the action choices on the dummy variable “Signal” which takes one if thesignal is good. The coefficient on the dummy variable captures the contrast between thecooperation rate contingent on the good signal and that on bad signal, which is theretaliation intensity. To examine the experience effects on retaliation intensities acrossrepeated games and with-in a repeated game, we add the cross-product terms with “RG2”,“RG3” and “First 14 Rd.” in the set of the explanatory variables. Again, the regression
model is a fixed-effect model in which the individual heterogeneity in tendencies to takecooperative choices is controlled by individual fixed effects.
[TABLE A. 4 HERE]
Table A. 4 displays the regression results. None of the coefficient on the joint effectof “Signal” and repeated games (“RG2” and “RG3”) is significantly different from zeroin both treatments, implying that the retaliation intensities do not differ either in thesecond repeated games or in third repeated games from that in the first repeated games.

51

Furthermore, the coefficient on the cross-product of “Signal” and “RG2” does not differ
from that on the cross-product of “Signal” and “RG3” in both treatments (p = 0.702 forthe high accuracy treatment, and p = 0.672 for the low accuracy treatment). These results
jointly indicate that the retaliation intensities are stable across repeated games.
As to the with-in repeated game difference, only the coefficient on the joint effect of“Signal” and “First 14 Rd.” in the high accuracy treatment significantly differs from zero,
although the size of the effect is approximately 5%. Our subjects tend to rely on stronger
retaliation intensities as the rounds proceed in a repeated game, however, the difference
is not remarkably large.
Overall, the results here indicate that the retaliation intensities do not change
remarkably as our subjects gain experience.
Table A. 1:
Fixed-effect model regression results on the experience effect on overall
cooperation rates
RG2
RG3
First 14 Rd.

p = 0.9
0.029 (0.031)
0.105*** (0.031)
0.062*** (0.014)

p = 0.6-0.071** (0.027)-0.118*** (0.029)
0.080*** (0.015)

Observations
R2

8,864
0.021

9,144
0.025

Notes: Cluster-robust (individual-level) standard errors in parenthesis. *p < 0.1, **p < 0.05, ***p
< 0.01. The coefficient on “RG3” is significantly larger than that on “RG2” in the high accuracy
treatment (F-test, p = 0.016). The coefficient on “RG3” is significantly smaller than that on “RG2”in the low accuracy treatment (F-test, p = 0.027).

Table A. 2:
Fixed effect model regression results on the experience effect on cooperation rates
contingent on good signals
RG2
RG3
First 14 Rd.

p = 0.90.014 (0.016)
0.076*** (0.021)
0.018 (0.011)

p = 0.6-0.055* (0.029)-0.104*** (0.034)
0.073*** (0.018)

Observations
R2

5,453
0.005

4,265
0.021

Notes: Cluster-robust (individual-level) standard errors in parenthesis. *p < 0.1, **p < 0.05, ***p
< 0.01.

52

Table A. 3:
Fixed effect model regression results on the experience effect on cooperation rates
contingent on bad signals
RG2
RG3
First 14 Rd.

p = 0.9
0.025 (0.030)
0.052 (0.036)
0.061*** (0.018)

p = 0.6-0.070** (0.030)-0.123*** (0.029)
0.070*** (0.016)

Observations
R2

3,087
0.009

4,555
0.027

Notes: Cluster-robust (individual-level) standard errors in parenthesis. *p < 0.1, **p < 0.05, ***p
< 0.01.

Table A. 4;
Fixed-effect model regression results on the experience effect on retaliation
intensities
SignalSignal: RG2
Signal: RG3
Signal: First 14 Rd.

p = 0.9
0.392*** (0.030)-0.015 (0.027)
0.007 (0.038)-0.048** (0.019)

p = 0.6
0.118*** (0.025)
0.020 (0.022)
0.034 (0.027)
0.005 (0.018)

RG2
RG3
First 14 Rd.

0.029 (0.034)
0.070* (0.039)
0.068*** (0.018)-0.077** (0.030)-0.130*** (0.029)
0.072*** (0.016)

Observations
R2

8,540
0.021

8,820
0.025

Notes: Cluster-robust (individual-level) standard errors in parenthesis. *p < 0.1, **p < 0.05, ***p
< 0.01. The coefficient on the cross-product term “Signal: RG2” is not significantly different fromthat on the cross-product term “Signal: RG3” in the high accuracy treatment (F-test, p = 0.702).
The coefficient on the cross-product term “Signal: RG2” is not significantly different from thaton the cross-product term “Signal: RG3” in the low accuracy treatment (F-test, p = 0.672).

Appendix 3: Derivation of Likelihood
The likelihood function in SFEM frameworks is derived as follows. The choiceprobability of subject i employing strategy s at round r of the repeated game k ,given the history of her past choices and signals obtained from the opponent up to theround, is defined as(A. 1)

Pikr ( s )1,1  exp(1/ J )if the observed choice is matched with the predicted choice by strategy s given thehistory up to the round. Otherwise, the choice is classified as implementation error and53

the choice probability is(A. 2)

Pikr ( s )1,1  exp(1/ J )where J captures the probability of the implementation error.
The likelihood of subject i employing strategy s is

Pi ( s )  P (s) .
ikr

k

r

Now in the SFEM framework, the likelihood of subject i over all strategies is a finite
mixture of Pi ( s ) over the entire strategy set. We denote the frequency of occurrence ofstrategy s by P( s ) . Then the log likelihood of the MLE is

LH

§

·

¦ ln ¨© ¦ P(s) P (s) ¸¹ .
iisThe choice probabilities in (A. 1) and (A. 2) are defined over deterministic strategies.
Since our list of strategies considered in our SEFM (Table 7) includes stochastic strategies,
the choice probabilities should be extended to cover stochastic cases. Following
Fudenberg, Rand, and Dreber (2012), the choice probabilities (A. 1) and (A. 2) areextended to cover stochastic strategies as follows,(A. 3)§·§·
1
1
Pikr ( s ) sikr ¨¸  (1  sikr ) ¨¸© 1  exp(1/ J ) ¹© 1  exp(1/ J ) ¹
if the observed choice is A,(A. 4)§·§·
1
1
Pikr ( s ) (1  sikr ) ¨¸  sikr ¨¸© 1  exp(1/ J ) ¹© 1  exp(1/ J ) ¹
if the observed choice is B,

where sikr is the probability of playing A in the stochastic strategy s given the history
up to the round. Observe that the new formulations of the choice probabilities (A. 3) and(A. 4) are reduced to the previous definition (A. 1) and (A. 2) when sikr takes either one
or zero as deterministic choices.
The standard error of the MLE is computed through a cluster-bootstrap (subjectlevel) with 100 resamples, which is also used to perform the hypothesis tests presented in
Section 8.

54

Appendix 4: Robustness Checks of Our Strategy Estimation
In this Appendix we discuss the robustness of the SFEM estimates. In the main text
we use all the three repeated games of each treatment in our estimation. Here wedemonstrates that the estimation results almost show little changes even only using thefinal two repeated games in each treatment (Table A. 5 and A. 6).
The mean retaliation intensity in the high accuracy treatment decreases slightly inthe final 2 repeated games, which is closer to the value implied by the g-TFT equilibria(0.235), which might suggest that our subjects learn optimal retaliation intensities byexperience in the high accuracy treatment.

Table A. 5
Aggregated estimates for high accuracy case (p = 0.9)
All

Final 2

Cooperative Strategies

83.9%

88.3%

Family of g-TFT (including zero RI strategies)

76.8%

75.2%

Equilibrium RI

5.6%

7.6%

Stronger RI

54.4%

49.0%

Weaker RI (including zero RI strategies)

17.7%

20.8%

Mean RI conditional on g-TFT

0.426

0.360

21.4%

21.6%

(including zero RI strategies)
Lenient Strategies

Table A. 6
Aggregated estimates for low accuracy case (p = 0.6)
All

Final 2

Cooperative strategies

34.7%

37.6%

Family of g-TFT (including zero RI strategies)

96.2%

96.5%

Equilibrium RI

2.7%

2.4%

Stronger RI

0.0%

0.0%

Weaker RI (including zero RI strategies)

93.5%

94.0%

Mean RI conditional on g-TFT

0.148

0.135

(including zero RI strategies)

55

3.7%

Lenient strategies

2.5%

Appendix 5: Proof of Theorem 2
The proof of the “if” part is straightforward from (7), (8), (9), (10), (11), and (12).
Fix an arbitraryp  ( p,1) . From continuity, we can assume without loss ofgenerality that1  r ( a; p ) z r (b; p ) .

Suppose p ! pˆ , i.e.,r ( a; p )  r (b; p ) ! w( p ) ! 0 .

This, along with w( p ) {Yand c(a; p ) t 0 , implies the inequality of (8), andG (2 p  1) Z

therefore,r ( a; p )1  H ( p ) and r (b; p )  1  H ( p ) .

Hence, from (9) and (10), either r (b; p ) H ( p ) orY  c(b; p )G Z (2 p  1){r (a; p )  r (b; p )} ,which implies (11), wherer (b; p ) z H ( p )holds becauser ( a; p ) 1  H ( p )and1  r ( a; p ) z r (b; p ) .

Next, suppose p  pˆ , i.e.,r (a; p )  r (b; p )  w( p ) .

This, along with w( p ) {Yand c(b; p ) t 0 , implies the inequality of (9), andG (2 p  1) Z

therefore,r (b; p )H ( p ) and r (a; p ) ! H ( p ) .

Hence, from (7) and (8), either r (a; p ) 1  H ( p ) orY  c ( a; p )G Z (2 p  1){r (a; p )  r (b; p )} ,which implies (12), where r (a; p ) z 1  H ( p ) holds because of r (b; p ) H ( p ) and1  r ( a; p ) z r (b; p ) .

Q.E.D.

56

Appendix 6: Experimental Instruction(October 6, 2006, Translation from Japanese into English)
Please make sure all the contents are in your envelope. The envelop should have thefollowing items.
1. Pen - 1
2. Instruction – 1 copy
3. Printed computer screen images – 1 copy
4. Bank transfer form – 1 sheet
5. Scratch paper – 1 sheet
If you have any missing item, please raise your hand quietly. We will collect the items at theend of the whole experiments, except for the scratch paper which you could keep with you.
Please look at the instruction (this material). Your will be asked to make decisions on acomputer terminal. You will earn “points” according to your performance in the experiments. The
points will be converted into monetary rewards at the exchange rate of 0.6 yen per point, which
will be paid in addition to the participation fee (1,500 yen). The total amount of money you will
receive from the experiments is,
The number of points earned × 0.6 yen + participation fee of 1,500 yen.
Any communication with other participants (i.e. conversation, exchange of signals) is notallowed during the experiments, otherwise you may be asked to leave the experiments.
Furthermore, you are not allowed to leave in the middle of the experiments unless a experimenter
allows or askes to do so. Please turn off your cell phones during the experiments.
Outline of Experiments
We will conduct 6 experiments across 2 sessions. Each session includes 3 experiments with
one practice experiment preceding to them. The 6 experiments are independent of each other; therecords of one experiment is not transferred to the other experiments. The experiments is
conducted via computer network. You are asked to make decisions in a computer terminal andinteract with other participants through the computer network.
All of the participants will be divided into pairs in each experiment. The pairs are maderandomly by the computer.
Each experiment consists of several rounds (i.e. Round 1, Round 2, Round 3 and so forth).
Later we will explain the rule that decides the number of rounds conducted in an experiment. In
each round, you are asked to choose one from two alternatives, which will be also explained below.
Those who have any question, please raise your hand quietly.
Decision Making
You will be asked to choose either A or B in each round. Your partner will also asked tochoose either A or B. Please look at the table.

The table summarizes the points you and your partner earn according to the combination of
the choices made by the two players. The characters in the left column marked by red color
indicates your choice, which is either A or B. The characters in the top row marked by light blue

57

color indicate the choice of your partner, which is also either A or B. In each cell, the numbers in
red on the left side indicate the points you earn, and the numbers in light blue on the right side
indicate the points your partner earns.
If both you and your partner select A,
both of you and your partner earns 60 points.
If you select A and your partner selects B,
you earn 5 points, and your partner earns 70 points.
If you select B and your partner selects A,
you earn 70 points, and your partner earns 5 points.
If both you and your partner select B,
both of you and your partner earns 15 points.
Please look at the table carefully, and ensure that you understand how the points will beawarded to you and your partner according to the choices made by the two players. Your earning
does not only depend on your choice, but also depends on the choice of the partner. Similarly,
your partner’s earning also depends on your choice as well as on her own choice.
Those who have any questions, please raise your hand quietly.

Session 1
Session 1 consists of three experiments, Experiment 1, Experiment 2, and Experiment 3
respectively. The three experiments follow the identical rules and will be conducted consecutively.
Observable Information
You are not allowed to observe whether your partner selected A or B directly. However, youwill receive signal a or signal b which has information on your partner’s choice. According to thefollowing rules, the computer determines stochastically whether signal a or signal b appears toyou;
If your partner selects A, you will observe,
signal a at the probability of 90% and signal b at the probability of 10%.
If your partner selects B, you will observe,
signal b at the probability of 90% and signal a at the probability of 10%.
In the same way, your partner will not know whether you selected A or B. However, yourpartner will also observe signal a or signal b which have information on your choice. According
to the following rules, the computer determines stochastically whether signal a or signal b appears
to your partner;
If you selects A, your partner will observe,
signal a at the probability of 90% and signal b at the probability of 10%.
If you select B, your partner will observe,
signal b at the probability of 90% and signal a at the probability of 10%.
The signal you receive and the signal your partner receives are decided independently andhave no correlation. Furthermore, the computer determines the signals independently in eachround.
We refer the stochastic rules for this signal generating process as signaling with 90%
accuracy.
Those who have any question, please raise your hand quietly.
Number of Rounds
The number of rounds in each experiment will be determined randomly. At the end of eachround, the computer will randomly select a number from 1 to 30 without replacement, so there is
1/30 chance for any number being selected by the computer. The number selected by the computeris uniformly applied to all participants.

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The experiment will be terminated when the number 30 is selected by chance.
The experiment will continue if any number other than 30 is selected. However, you will be
only noticed that the number other that 30 is selected, instead of the specific number selected by
the computer. Then you will move on to the next round, and are again asked to make a decision
faced with the same partner.
The probability that the experiment is terminated in a round remains the same, which is 1/30,
regardless of the number of the rounds (Round 1, Round 2, Round 3, and so forth). However, themaximum possible number of rounds in an experiment is experimentally controlled, which is 98.
When Experiment 1 is terminated, you proceed to Experiment 2 and you will be randomly
paired with a new partner. When Experiment 2 is terminated, you proceed to Experiment 3 andagain you will be randomly paired with a new partner. Session 1 will be over when Experiment 3
is terminated.
Those who have any questions, please raise your hand quietly.
Description of Screens and Operations for Computers
Please look at the booklet with printed computer screen images.
Please look at Screen 1 and Screen 2. Screen 1 displays the screen which will be presented
to you at the decision phases. Screen 2 is the screen which will be presented to your partner at thedecision phases. Please look at the top left portion of each screen, which indicates that the current
round is Round 4. The left portion of Screen 1 displays the information available to you up to theround. The left portion of Screen 2 presented to your partner displays the information available
to her up to the round.
You are asked to select either “A” or “B” in the bottom right portion of the screen by mouseclicking. Then, the selection will be confirmed by clicking the “OK” button right below thealternatives.
Next, please look at Screens 3 and Screen 4. Screen 3 is the screen that presents the results
to you. Screen 4 is the screen that presents the results to your partner. The screens capture thesituation that, at Round 4, both of you and your partner chose A. Screen 3 presents to you that, atRound 4, “your partner’s signal (accuracy: 90%) is b,” indicating to you that the signal you
observe on the partner’s choice is “b”. On the other hand, Screen 4 presents to your partner that,at Round 4, “your partner’s signal (accuracy: 90%) is a”, indicating to your partner that the signal
your partner observe on your choice is “a”. Recall that your partner will observe signal a with theprobability of 90%, and observe signal b with the probability of 10% when you choose “A”.
Then we move on to the Lottery screens. Please turn the page and look at Screen 5 and Screen
6, which display the lottery. A number from 1 to 30 will be randomly selected with the identicalprobability of occurrence, which is 1/30. Then the cells turn to green according to the number
selected. If the 30 is selected, the cell numbered 30 will turn to green and the message below
explains that the current experiment is terminated.
Otherwise, Screen 5 is presented in which all the cells numbered from 1 to 29 turn to green
at once (you do not know which number is selected specifically), and the message below explainsthat the experiment continues with the same partner. Screen 6 is presented when the number 30 isselected and the cell numbered 30 turns to green, indicating that the current experiment isterminated at the round. Again, please make sure that the experiment is terminated when the cellnumbered 30 turns to green.
Finally, please look at Screen 7. This screen is presented at the end of experiment. The screen
displays the total number of points you earned in that experiment, the average number of points
per round, the total number of points your partner earned, and the average number of points perround of your partner. Then you will be rematched with a new partner and move on to the next
experiment.
Those who have any question, please raise your hand quietly.

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Session 2
Please look at page 6 of your instruction. In session 2, you will participate in threeexperiments, namely, Experiment 4, Experiment 5, and Experiment 6 with a practice experimentpreceding to them.
The three experiments follow identical rules and will be conducted consecutively. Session 2
proceed similarly to Session 1, however, the signal accuracy of Session 2 is different from that of
Session 1. Except for the signal accuracy, the two sessions are identical.
Observable Information
You are not allowed to observe whether your partner selected A or B directly. However, youwill receive signal a or signal b which have information on your partner’s choice. According to
the following rules, the computer determines stochastically whether signal a or signal b appears
to you;If your partner selects A, you will observe,
signal a at the probability of 60% and signal b at the probability of 40%.
If your partner selects B, you will observe,
signal b at the probability of 60% and signal a at the probability of 40%.
In the same way, your partner will not know whether you selected A or B. However, yourpartner will also observe signal a or signal b which has information on your choice. According to
the following rules, the computer determines stochastically whether signal a or signal b appears
to your partner;
If you selects A, your partner will observe,
signal a at the probability of 60% and signal b at the probability of 40%.
If you select B, your partner will observe,
signal b at the probability of 60% and signal a at the probability of 40%.
The signal you receive and the signal your partner receives are decided independently andhave no correlation. Furthermore, the computer determines the signals independently in eachround.
We refer the stochastic rules for the signal generating process as signaling with 60%
accuracy.
Those who have any question, please raise your hand quietly.
Description of Screens and Operations for Computers
Please look at the booklet with printed computer screen images.
Please look at Screen 8 and Screen 9. Screen 8 displays the screen which will be presented
to you at the decision phases. The left portion of Screen 8 displays the information available toyou up to the round. Please make sure the current signal accuracy with the message “Signal
accuracy for your partner's selection: 60%.” Screen 9 is the screen which will be presented to yourpartner at the decision phases. The left portion of Screen 9 presented to your partner displays theinformation available to her up to the round.
Please look at Screen 10 and Screen 11 on page 6. Screen 10 is the screen that presents theresults to you. Screen 11 is the screen that presents the results to your partner. The screens capture
the situation that, at Round 4, both of you and your partner chose A, however, you observed signalb, and your partner observed signal a. Please make sure that the bottom right portion of Screen 10
and Screen 11 displays the signal you or your partner observed respectively.
The choices you made and the signals you observe about your partner’s choices are only
available to you, and are not available to the partner. Please make sure this point in Screen 10 and
Screen 11.
The screens for the lottery on continuation of the experiment is identical to the case of
Session 1, which shows numbers from 1 to 30. Please refer back to Screen 5 and Screen 6. The

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results screen at the end of experiment is also identical to that in Session 1 (Screen 7).
Those who have any question, please raise your hand quietly.
Now, all the process of the experiments are completed, and all the points awarded to everyone
are recorded on the computer.
Please answer the questionnaire which will be distributed now.
Take out the bank transfer form in the envelop and fill out it accurately, otherwise we are notable to process the payment correctly for you.
Those who have any question, please raise your hand quietly.
Please make sure that you fill out the questionnaire and the bank transfer form correctly.
Those who have any question, please raise your hand quietly.
Please put all the documents in the envelope. Please leave the pen and the ink pad on thedesk. Make sure you take all your belongings with you when you leave.
Please do not to disclose any details regarding the experiments to anyone until Saturday.
Thank you very much for your participation. Please follow the instructions by the experimenters
to leave the room.

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2. Computer Screen Images(October 6, 2006, Translation from Japanese into English)
Screen 1: Your Selection Screen

Screen 2: The Selection Screen of Your Partner

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Screen 3: Your Results Screen

Screen 4: The Results Screen of Your Partner

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Screen 5: Lottery (experiment continues)

Screen 6: Lottery (experiment is over)64

Screen 7: The Results Screen

Screen 8: Your Selection Screen

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Screen 9: The Selection Screen of Your Partner

Screen 10: Your Results Screen

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Screen 11: The Results Screen of Your Partner